Browse Function problems Studdy Solved!

Problem 2401

Consider the equation below. (If an answer does not exist, enter DNE.) $f(x)=6 \cos ^{2}(x)-12 \sin (x), \quad 0 \leq x \leq 2 \pi$ (a) Find the interval on which $f$ is increasing. (Enter your answer using interval notation.) $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
Find the interval on which $f$ is decreasing. (Enter your answer using interval notation.) $\left(0, \frac{\pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right)$ (b) Find the local minimum and maximum values of $f$ . local minimum value $\quad-12$ local maximum value 12 (c) Find the inflection points. (Order your answers from smallest to largest $x$ , then from smallest to largest $y$ .) $\begin{array}{l} (x, y)=\left(\frac{\pi}{6}, \frac{3}{2}\right) \\ (x, y)=\left(\frac{5 \pi}{6}, \frac{3}{2}\right) \end{array}$
Find the interval on which $f$ is concave up. (Enter your answer using interval notation.) $\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, \frac{3 \pi}{2}\right)$
Find the interval on which $f$ is concave down. (Enter your answer using interval notation.)

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Problem 2402

The function $f(x)$ is shown in the table. \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & -4 & -2 & 0 & 2 & 4 \\ \hline $f(x)$ & -18 & -12 & -6 & 0 & 6 \\ \hline \end{tabular}
The function $g(x)$ is represented by the equation $g(x)=2(x-2)^{2}$ . Use the drop-down menus to compare the $y$ -intercepts of $f(x)$ and $g(x)$ .
Click the arrows to choose an answer from each menu. The $y$ -intercept of $f(x)$ is choose... and the $y$ -intercept of $g(x)$ is
Choose... The $y$ -intercept of $f(x)$ is choose... the $y$ -intercept of $g(x)$ .

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Problem 2403

If $f(x)=\sin \left(4 x+\frac{\pi}{2}\right)$ , find $f^{\prime}\left(\frac{\pi}{4}\right)$ . 0 4 $-4$ 1

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Problem 2404

Level J CC. 1 Evaluate a linear function LNV Analytios You have prizes t
Use the function $y=14 t$ to find the value of $y$ when $t=8$ . Video $y=$ $\square$ Submit

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Problem 2405

Read the following description of a relationship:
Hakim received a large bag of candy for his birthday. He ate 7 pieces on his birthday and ate 3 pieces each day after.
Let $d$ represent the number of days since Hakim's birthday and $p$ represent the number of pieces of candy Hakim has eaten.
Use the function $p=3 d+7$ to find the value of $p$ when $d=1$ . $p=$ Submit

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Problem 2406

Match each function with the ordered pairs in the function tables.
15. $y=\frac{1}{2} x-5$ 4. \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & 2 & 4 & 6 & 8 & 10 \\ \hline $y$ & -4 & -3 & -2 & -1 & 0 \\ \hline \end{tabular} $\qquad$ 16. $y=4 x+3$ B. \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & 3 & 6 & 9 & 12 & 15 \\ \hline $y$ & -1 & -3 & -5 & -7 & -9 \\ \hline \end{tabular} $\qquad$ 17. $y=-\frac{2}{3} x+1$ $\qquad$ 18. $y=2 x+6$ c. \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & 1 & 2 & 3 & 4 & 5 \\ \hline $y$ & 7 & 11 & 15 & 19 & 23 \\ \hline \end{tabular} D. \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & 3 & 4 & 5 & 6 & 7 \\ \hline $y$ & 12 & 14 & 16 & 18 & 20 \\ \hline \end{tabular}

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Problem 2407

You deposit $\$ 3000$ in an account earning $7 \%$ interest compounded continuously. The amount of money in the account after $t$ years is given by $A(t)=3000 e^{0.07 t}$ .
How much will you have in the account in 5 years? \\qquad$ Round your answer to 2 decimal places.
How long will it be until you have $\$ 17600$ in the account? $\square$ years. Round your answer to 2 decimal places.
How long does it take for the money in the account to double? $\square$ years. Round your answer to 2 decimal places.

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Problem 2408

**Which of the following statements is true? I. $\quad \sin ^{-1}\left(\sin \left(\frac{7 \pi}{4}\right)\right)=\frac{7 \pi}{4}$ II. $\cos ^{-1}\left(\cos \left(\frac{2 \pi}{3}\right)\right)=\frac{2 \pi}{3}$ III. $\tan ^{-1}\left(\tan \left(-\frac{\pi}{6}\right)\right)=-\frac{\pi}{6}$

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Problem 2409

V EXAMPLE 2 Where are each of the following functions discontinuous? (a) $f(x)=\frac{x^{2}-x-2}{x-2}$ (b) $f(x)=\left\{\begin{array}{ll}\frac{1}{x^{2}} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{array}\right.$ (c) $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-x-2}{x-2} & \text { if } x \neq 2 \\ 1 & \text { if } x=2\end{array}\right.$ (d) $f(x)=\llbracket x \rrbracket$

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Problem 2410

State the range of the following functions: a) $f(x)=\arccos x$ b) $f(x)=\sin x$ c) $f(x)=\tan ^{-1} x$

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Problem 2411

The position of an object at t seconds is given in feet by $f(t)=3 t^{2}+4 t^{-1}+\sqrt{t}$
What is the object's velocity at 4 seconds?

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Problem 2412

The acceleration of an object at $t$ seconds is given in feet per second per second by $f(t)=(t+2)(t+3)$
At 1 second the object's velocity is 9 feet per second. What is the object's velocity at 3 seconds?

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Problem 2413

Identify the transformation from the graph of $f(x)=x^{2}$ to the graph of $g(x)$ . Then $\operatorname{graph} f(x)$ and $g(x)$ on the same coordinate plane.
17. $g(x)=x^{2}-7$
18. $g(x)=x^{2}+10$

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Problem 2414

(D) The cost. in dollars, of producing $s$ desks has tripled
2. Determine if each of the following relationships can be represented by a function 1 where $y=1(x)$ . Gives reason for your answer. a. The base of a triangle is 4 inches, the height is $x$ inches, and the area is $y$ square inches b. A set of ordered pairs where $x=$ any integer and $y=$ the cube root of $z$ c. \begin{tabular}{c|cccccc} $x$ & 3 & 1 & 4 & 1 & 5 & 9 \\ \hline $f(x)$ & 2 & 7 & 1 & 8 & 2 & 8 \end{tabular} d. $\{(1,5),(3,4),(5,0),(7,16),(9,25)\}$

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Problem 2415

3. A function $f(x)$ is defined as $f(x)=3 x-10$ . a) Given that the range of $f(x)$ is $5<f(x)<50$ , find the domain of $f(x)$ . (2 marks) b) Find $f(f(10))$ . (2 marks)

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Problem 2416

$\begin{array}{l}f(t)=\left\{\begin{array}{ll}t^{2}-5 t & , \quad t \leq-10 \\ t+19 & ,\end{array},-10<t<-2\right. \\ \frac{t^{3}}{t+9}\end{array}, \quad t \geq-2$ .

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Problem 2417

The half-life of a certain type of soft drink is 7 hours. If a person drinks 51 millifters of this drink, the formula $y=51(0.5)^{\frac{1}{7}}$ tells the amount of the drink left in the person's system after thours. How m the soft drink is in the person's system after 8 hours?
The person's system contains $\square$ $\square$ of the soft drink. (Type an integer or decimal rounded to the nearest tenth as needed.)

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Problem 2418

Suppose that the function $f$ is defined, for all real numbers, as follows. $f(x)=\left\{\begin{array}{ll} -x+2 & \text { if } x \leq 1 \\ 5 x-2 & \text { if } x>1 \end{array}\right.$
Graph the function $f$ . Then determine whether or not the function is continuous.

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Problem 2419

Question 4 In 1990, the value of a home was $\$ 170,000$ . Since then, its value has increased 5\% per year. 5 Numeric 1 point Question 4 Part $A$
What was the approximate value of the home in the year 1993? (Round to the nearest hundred.)
Type your answer.
6 Fill in the Blank 1 point
Question 4 Part B Write an equation, in function notation, to represent the value of the home as a function of time in years since 1990, $t$ . $v(t)=$ type your answer... 7 Fill in the Blank 3 points
Question 4 Part $C$ Will the value of the home be more than $\$ 500,000$ in 2020 (assuming that the trend continues)?
Show/explain your reasoning. choose your answer. $\square$ because the value of the home will be \ $type your answer.. which is$ \square$ choose your answer.

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Problem 2420

Question 6 Here is the graph of an exponential function $f$ . Find an equation defining $f$ . $f(x)=$ $\square$ type your answer...

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Problem 2421

Question 26 (Mandatory) (1 point) The function $h(t)=-5 t^{2}+7 t+12$ , where $h(t)$ is the height in metres and $t$ is the time in seconds, models the height of a ball thrown from a balcony to the ground below. What is the maximum height reached by the ball?

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Problem 2422

Where is the graph of $f(x)=4\lfloor x-3\rfloor+2$ discontinuous? all real numbers all integers only at multiples of 3 only at multiples of 4

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Problem 2423

A skate shop rents roller skates as shown in the table below.
Roller Skate Rentals \begin{tabular}{|c|c|} \hline Time & Cost \\ \hline up to 60 minutes & $\$ 5$ \\ \hline up to 2 hours & $\$ 10$ \\ \hline up to 5 hours & $\$ 20$ \\ \hline daily & $\$ 25$ \\ \hline \hline \end{tabular}
Which graph and function model this situation, where $c$ is the cost, in dollars, of renting skates, and $t$ is the number of hours in a single day that the skates are rented? $c(t)=\left\{\begin{array}{r}5 \text { if } 0<t \leq 2 \\ 10 \text { if } 2<t \leq 4 \\ 20 \text { if } 4<t \leq 6 \\ 25 \text { if } 6<t \leq 8\end{array}\right.$

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Problem 2424

Ontarionool
MHF4U Test \#3: Chapters 6-7 Name: $\qquad$ Part A - Multiple Choice [K/U - 15 marks]
1. The function $f(x)=\left(\frac{1}{4}\right)^{x}$ passes through the point a. $(-1,4)$ b. $(1,-4)$ c. $(-1,-4)$
Date: $\qquad$
2. The $x$ -intercept of the function $y=6^{x}$ is a. 1 b. 6 c. 0 d. does not exist
3. The exponential function for the following data set is a. $y=9^{x}$ b. $y=\left(\frac{1}{9}\right)^{x}$ c. $y=3^{x}$ d. $y=\left(\frac{1}{3}\right)^{x}$ \begin{tabular}{|c|c|} \hline $\boldsymbol{x}$ & $\boldsymbol{y}$ \\ \hline 2 & 9 \\ \hline 3 & 27 \\ \hline 4 & 81 \\ \hline 5 & 243 \\ \hline \end{tabular}
4. Another way of writing $8^{3}=512$ is a. $\quad \log _{8} 512=3$ b. $\log _{3} 512=8$ c. $\log _{512} 8=3$ d. $\log _{8} 3=512$
5. Another way of writing $a=\log _{2} \frac{1}{16}$ is a. $\quad a^{2}=\frac{1}{16}$ b. $\left(\frac{1}{16}\right)^{2}=2$ c. $a^{\frac{1}{16}}=2$ d. $2^{a}=\frac{1}{16}$
6. Evaluate $\log _{3} 27$ . a. 0 b. 1 c. 2 d. 3
7. The equation of the vertical asymptote for the function $y=3 \log (x+4)$ is a. $x=-4$ b. $x=3$ c. $x=4$ d. $x=0$
8. The $x$ -intercept of the function $y=4 \log (x+6)$ is a. 4 b. -5 c. 5 d. 6
9. Express $\sqrt[4]{1024}$ as a power with a base of 2 . a. $2^{10}$ b. $2^{\frac{2}{5}}$ c. $2^{\frac{5}{2}}$ d. $2^{\frac{3}{4}}$
10. Express $17^{4}$ as a power with a base of 2 . a. $\log _{2} 17^{4}$ b. $2^{\frac{\log 2}{\log 17}}$ c. $\log _{2} 17 \times \log _{2} 4$ d. $2^{\frac{4 \log 17}{\log 2}}$

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Problem 2425

$f(x)=\left\{\begin{array}{rrc} -10+x^{2} & \text { if } & -4 \leq x<4 \\ 12-x & \text { if } & x \geq 4 \end{array}\right.$
Graph the function $f$ . Then determine whether or

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Problem 2426

Jimmy receives his first 3 video games free then pays $\$ 50$ for each game after that.
Is the amount of money he spends on video games proportional to the number of games he owns?
Choose 1 answer: (A) Yes (B) No

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Problem 2427

nt/3354527/25279753/66e087c9cbf4583acf59eeb0809cf419
The graph of $y=f(x)$ is shown below. What are all of the real solutions of $f(x)=0$ ?

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Problem 2428

Name: $\qquad$ siya $\qquad$ a oa
Part A - Multiple Choice [K/U - 15 marks]
1. The function $f(x)=\left(\frac{1}{4}\right)^{x}$ passes through the point a. $(-1,4)$ b. $(1,-4)$ c. $(-1,-4)$ d. $(1,4)$
2. The $x$ -intercept of the function $y=6^{x}$ is d. does not exist a. 1 b. 6 c. 0 \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 2 & 9 \\ \hline 3 & 27 \\ \hline 4 & 81 \\ \hline 5 & 243 \\ \hline \end{tabular}
3. The exponential function for the following data set is a. $y=9^{x}$ b. $y=\left(\frac{1}{9}\right)^{x}$ c. $y=3^{x}$ d. $y=\left(\frac{1}{3}\right)^{x}$ c. $\log _{512} 8=3$ d. $\log _{8} 3=512$
4. Another way of writing $8^{3}=512$ is a. $\log _{8} 512=3$ b. $\log _{3} 512=8$ (4). $\log _{512} 8=3$ $\qquad$ $\qquad$ (5it $\qquad$

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Problem 2429

Which statement explains why the value of $\lfloor 2.4\rfloor$ is 2 but the value of $\lfloor-2.4\rfloor$ is -3 ? because 2 is the greatest integer not greater than 2.4 , and -3 is the greatest integer not greater than -2.4 because 2.4 rounds to 2 , and -2.4 rounds to -3 because 2.4 is positive, and -2.4 is negative because 2 is the least integer greater than 2.4, and -3 is the least integer greater than -2.4

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Problem 2430

The exponential function given by $\mathrm{H}(\mathrm{t})=80,034.64(1.0488)^{\mathrm{t}}$ , where t is the number of years after 2006, can be used to project the number of centenarians in a certain country. Use this function to project the centenarian population in this country in 2011 and in 2038.
The centenarian population in 2011 is approximately $\square$ (Round to the nearest whole number.)

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Problem 2431

1. Lifeguard certification requires a lifeguard to swim underwater and stay below 3 m of depth for at least 5.5 seconds. Megan's underwater swim is described below. Did Megan get her lifeguarding certification? Justify your answer. $y=\left\{\begin{array}{cc} a x^{2}, & 0 \leq x \leq 2 \\ -5 & , 2 \leq x \leq 6 \\ a x+b, & 6 \leq x \leq 8 \end{array}\right\}$ 1) $\qquad$ 2) $y=a x+b$

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Problem 2432

Find the function $\lambda$ for the Heisenberg group representation where $V=V_{\lambda}$ and $\lambda$ is not linear.

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Problem 2433

Compare $x - y$ and $(x + y)^{2}$ given $x > 0 > y$ .

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Problem 2434

Find the domain and range of the function $f(x)=4(2^{-2x+7})-5$ .

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Problem 2435

Identify the transformation of the function $f(x)=-3\left(5^{2 x-6}\right)+3$ : a) Right 3 units b) Down 6 units c) Stretch by 3 d) Reflection on $y$ -axis

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Problem 2436

Find the derivative $f^{\prime}(x)$ of $f(x)=|x|$ for $x>0$ , $x<0$ , and at $x=0$ . Explain why it's undefined at $0$ .

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Problem 2437

In a game, Andy scores 10% more than Bill, and Bill scores 10% less than Chris, who scored 700. Are Andy and Chris's scores equal? By what percent is the highest score greater than the lowest?

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Problem 2438

Given values of $f^{\prime \prime}(x)$ : at $x=-1$ is -4, $x=0$ is -1, $x=1$ is 2, $x=2$ is 5, $x=3$ is 8. What type of function is $f$ ? (A) linear (B) quadratic (C) cubic (D) fourth-degree (E) exponential

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Problem 2439

Find the value of $c$ such that $f(x)=x+\frac{c}{x}$ has a local minimum at $x=3$ .

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Problem 2440

Find the time $t$ (from 0 to 10) when the object with velocity $v(t)=t \cos t-\ln (t+2)$ reaches maximum speed.

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Problem 2441

Calculate the area under the curve $y=\frac{4}{1+x^{2}}$ from $x=-1$ to $x=1$ .

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Problem 2442

Find the time $t$ in $[0, 10]$ when the velocity $v(t)=t \cos t - \ln(t+2)$ is maximized. Options: A. 9.5 B. 5.1 C. 6.4 D. 7.6

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Problem 2443

In a game, Peter scored $x$ , Mary scored $y$ , and Susan scored $z$ . Show $z$ in terms of $x$ and $y$ .

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Problem 2444

In 2000, Mexico's population was 100 million, growing at $1.53 \%$ per year. When will it reach 109 million? Round to nearest year. a. 2006 b. 2008 c. 2005 d. 2007 (Select One)

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Problem 2445

Write a recursive rule for the function $y=18(8)^{x}$ . Choose one:
A $f(0)=8$ , $f(n)=18 \circ f(n-1)$ B $f(0)=18$ , $f(n)=18 \cdot f(n-1)$ C $f(0)=18$ , $f(n)=8 \cdot f(n-1)$ D $f(0)=8$ , $f(n)=8 \cdot f(n-1)$

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Problem 2446

Find the rate of change of sales $S(t)=10,000+2000 t-200 t^{2}$ and interpret $S^{\prime}(t)$ .

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Problem 2447

Find the equivalent expression for $\sin \left(\frac{\pi}{5}\right)$ from the options: A) $-\cos \left(\frac{\pi}{5}\right)$ B) $-\sin \left(\frac{\pi}{5}\right)$ C) $\cos \left(\frac{3 \pi}{10}\right)$ D) $\sin \left(\frac{7 \pi}{10}\right)$

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Problem 2448

Find the value of the constant $c$ in $f(x)=2x^3+3x^2+cx+8$ given $f$ intersects the $x$ -axis at $(-4,0)$ , $(\frac{1}{2},0)$ , and $(p,0)$ .

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Problem 2449

Find the derivative of the function $f(x)=6 \cdot \sqrt{x} \cdot(x-5)$ . What is $f^{\prime}(x)$ ?

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Problem 2450

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-3 \cdot \ln (4 \cdot x)$ and calculate $f^{\prime}(2)$ .

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Problem 2451

Find the derivative of $f(x)=\frac{\sqrt{x}-7}{\sqrt{x}+7}$ and calculate $f^{\prime}(5)$ .

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Problem 2452

Find the derivative $f'(x)$ of the function $f(x)=\sqrt{4x^2 + 4x + 4}$ and evaluate it at $x=4$ .

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Problem 2453

Calculate the derivative of $g(x)=(4x^{2}+5x) \cdot e^{x}$ .

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Problem 2454

Find values of $x$ where $f(x)=\begin{cases}x^{2}-1, & x<1 \\ x, & x \geq 1\end{cases}$ is discontinuous and explain why.

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Problem 2455

Find discontinuities of $f(x)=\left\{\begin{array}{ll}x^{2}-1, & x<1 \\ x, & x \geq 1\end{array}\right.$ and explain why.

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Problem 2456

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2+\frac{7}{x}+\frac{7}{x^{2}}$ .

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Problem 2457

Find the derivative of $R(j)=\left(\ln \left(j^{2}\right)\right)^{2}$ at $j=e$ .

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Problem 2458

Find the derivative $f^{\prime}(t)$ of the function $f(t)=(t^{2}+5t+3)(2t^{-2}+6t^{-5})$ .

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Problem 2459

Find the marginal cost for both businesses given their cost functions: $C_A(x)=200+25x+0.1x^2$ and $C_B(x)=400+80x+0.06x^2$ . For $x=500$ , which business has the lowest cost for the next tire?

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Problem 2460

Find the critical number of the function $f(x)=(8 \cdot x-7) \cdot e^{5 \cdot x}$ .

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Problem 2461

Find the length $L$ and width $W$ (with $W \leq L$ ) of a rectangle with perimeter 28 that maximizes area. What is the max area?

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Problem 2462

The demand function is $D(x)=967-25-x$ .
(a) Find the elasticity of demand. (b) Find the price where elasticity equals 1. (c) Determine the price for maximum revenue.

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Problem 2463

Projectile height is given by $h(t)=-16 \cdot t^{2}+256-t$ . Find: (a) average velocity for 3s, (b) speed & height at 6s, (c) max height & time, (d) acceleration at $t=5$ .

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Problem 2464

Approximate the integral $\int_{-1}^{2}(x^{4}-3 \cdot x^{2}+1) d x$ using left Riemann sums with 8 subdivisions. Round to three decimal places.

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Problem 2465

Find the total displacement of a ball with velocity $v(t)=90-32 \cdot t$ from $t=2$ to $t=5$ seconds.

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Problem 2466

Calculate the area between $y=\sqrt{x}$ and $y=0$ over the interval $[-1,1]$ .

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Problem 2467

Calculate the 5-unit moving average of the function $f(x) = x^3 - x$ .

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Problem 2468

Find the average rate of change of profit $P(x)=-400x^2+6800x-12000$ over $[6,6+h]$ for $h=1, 0.1, 0.01, 0.001, 0.0001$ . What do the results indicate?

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Problem 2469

A particle's position is given by $s=2 \cdot t^{3}-7 \cdot t^{2}-9 \cdot t+12$ . Find velocity, acceleration, and specific values.

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Problem 2470

Projectile height is given by $h(t)=-16 \cdot t^{2}+256 \cdot t$ . Find average velocity, speed, max height, and acceleration at $t=5$ .

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Problem 2471

Show that the limit $\operatorname{limit}_{x \rightarrow 2} \frac{|x-2|}{x-2}$ does not exist.

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Problem 2472

Find the speed of a satellite in a circular orbit just above Earth, given radius $6.4 \times 10^6 \mathrm{~m}$ and $g=9.8 \mathrm{~m/s}^2$ .

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Problem 2473

Find $\sqrt[3]{y}$ given that $(3,8)=(3, \sqrt{y})$ .

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Problem 2474

If $(x-1,11)=(8, y+3)$ , find $\sqrt{x+2y}$ . Options: (a) 5 (b) $\pm 5$ (c) $\sqrt{17}$ (d) 25

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Problem 2475

Which statements about temperature changes are true based on the relationship between $F$ and $C$ ? I. 1°F = $\frac{5}{9}$ °C II. 1°C = 1.8°F III. $\frac{5}{9}$ °F = 1°C A) I only B) II only

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Problem 2476

Find $f(x)=\frac{x-3}{x^{2}-3 x-4}$ for $x=-2, 0, 1, 3$ . Determine the domain.

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Problem 2477

Find $x^{5} y + 1$ if $(x+2, y) = (2, 3)$ . Options: (a) 3 (b) 2 (c) 0 (d) 1.

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Problem 2478

Find values of the functions:
1. $f(x)=\frac{x-3}{x^{2}-3 x-4}$ for $x=-2, 0, 1, 3$
2. $g\left(x^{2}\right)=4-\sqrt{x+9}$ for $x=-8, -5, 0, 7$

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Problem 2479

1. If $(x-1,11)=(8,y+3)$ , find $\sqrt{x+2y}$ . (a) 5 (b) $\pm 5$ (c) $\sqrt{17}$ (d) 25
2. If $(x+2,y)=(2,3)$ , find $x^5 y + 1$ . (a) 3 (b) 2 (c) 0 (d) 1
3. If $(3^x,\sqrt{y})=(1,4)$ , find $x+y$ . (a) 2 (b) 3 (c) 16 (d) 17
4. If $(a+2,63)=(-1,b^3-1)$ , find $\sqrt{a^2+b^2}$ . (a) 25 (b) 7 (c) 5 (d) $\pm 5$
5. If $(x-3,2^y)=(2,16)$ , find $(y,x)$ . (a) $(1,4)$ (b) $(5,4)$ (c) $(4,1)$ (d) $(4,5)$

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Problem 2480

Find the exponential function with an initial value of 3: $f(x)=\frac{1}{3}(9)^{x}$ or $f(x)=(3)^{x}$ .

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Problem 2481

Determine which statement is true for the function $f(x)=a\left(2^{x}\right)$ given the values of $a$ and $f(0)$ or $f(1)$ .

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Problem 2482

Find the growth factor of the function $f(x)=\frac{1}{5}\left(15^{x}\right)$ . Options: $\frac{1}{5}$ , $\frac{1}{3}$ , 5, 15.

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Problem 2483

Define the profit function $P(x)$ from cost $C(x)=56,000+37x$ and revenue $R(x)=41x$ . Then find $P(20000)$ .

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Problem 2484

Define profit $P(x)$ from cost $C(x)=60,000+45x$ and revenue $R(x)=50x$ . Find $P(19000)$ .

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Problem 2485

A machine costs R120,000 and depreciates at $9\%$ per year. Find its value after 5 years, new cost with $7\%$ inflation, and monthly deposits for R90,000 at $8.5\%$ interest.

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Problem 2486

Graph $g(x)=|x-2|$ and compare it to $f(x)=|x|$ . Describe the domain and range. Select the correct translation and range.

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Problem 2487

Graph $g(x)=-\frac{1}{4}|x|$ . Compare to $f(x)=|x|$ and describe domain and range. Choose the correct option.

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Problem 2488

Describe the transformations from $f(x)=|x|$ to $g(x)=2|x+1|-2$ and graph $g(x)$ . Choose the correct transformation option.

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Problem 2489

Identify the transformations from $f(x)=|x|$ to $g(x)=-|x+2|+3$ and choose the correct description.

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Problem 2490

Evaluate $g(2)$ for the piecewise function:
$g(x)=\left\{\begin{array}{ll} -x-1, & \text { if } x \leq-3 \\ -6, & \text { if }-3<x<1 \\ 4 x+1, & \text { if } x \geq 1 \end{array}\right.$
Options: A -3, B -6, C -9, D 9

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Problem 2491

24,000 fans attended last week. This week, three times that bought tickets, but one-sixth canceled. How many are attending? A. 48000 B. 54000 C. 60000 D. 72000

See Solution

Problem 2492

Is the transformation from $y=3^{x}$ to $y=5 \cdot 3^{-(x-2)}+9$ unique? List transformations to prove or disprove the claim.

See Solution

Problem 2493

Find the conversion factor from Wiffles to Inches using the table and fill in the missing values.

See Solution

Problem 2494

Calculate the integral $\int 2 x \cos \left(x^{2}-5\right) \, dx$ .

See Solution

Problem 2495

Maxwell bought skis for \ $350, paid \$110 down, got a \$30 discount, and his mom gave him half the balance. What expression finds what he owes? A.$ 350-110+30 \div 2 $B.$ 350-(110-30) \div 2 $C.$ [350-(110-30) \div 2] $D.$ [350-(110+30)] \div 2$

See Solution

Problem 2496

A health club charges \ $45 for the first month and \$35 for each additional month. Write the step function$ f(x)$ for costs.

See Solution

Problem 2497

Determine which statements are true based on the temperature conversion between Fahrenheit $F$ and Celsius $C$ :
1. 1°F = $\frac{5}{9}$ °C increase
2. 1°C = 1.8°F increase
3. $\frac{5}{9}$ °F = 1°C increase

See Solution

Problem 2498

Graph the piecewise function and identify its domain and range. $y=\left\{\begin{array}{ll} -2 x+1, & \text { if } x \leq-2 \\ 1, & \text { if } -2<x<1 \\ 2 x, & \text { if } x \geq 1 \end{array}\right.$ Options: A) Domain: all reals, Range: $y=-1, y \geq 2$ B) Domain: all reals, Range: $y=1, y \geq 2$ C) Domain: all reals, Range: $y=-1, y \geq 2$ D) Domain: all reals, Range: $y=1, y \geq 2$

See Solution

Problem 2499

Differentiate $y=\sqrt{a+x}$ .

See Solution

Problem 2500

Differentiate $y=\frac{1}{\sqrt{a+x^{2}}}$ with respect to $x$ .

See Solution

Function

Problem 2401

Problem 2402

Problem 2403

If f(x)=sin⁡(4x+π2)f(x)=\sin \left(4 x+\frac{\pi}{2}\right)f(x)=sin(4x+2π​), find f′(π4)f^{\prime}\left(\frac{\pi}{4}\right)f′(4π​). 0 4 −4-4−4 1

Problem 2404

Level J CC. 1 Evaluate a linear function LNV Analytios You have prizes tUse the function y=14ty=14 ty=14t to find the value of yyy when t=8t=8t=8. Video y=y=y= □\square□ Submit

Problem 2405

Problem 2406

Problem 2407

Problem 2408

Problem 2409

Problem 2410

State the range of the following functions: a) f(x)=arccos⁡xf(x)=\arccos xf(x)=arccosx b) f(x)=sin⁡xf(x)=\sin xf(x)=sinx c) f(x)=tan⁡−1xf(x)=\tan ^{-1} xf(x)=tan−1x

Problem 2411

The position of an object at t seconds is given in feet by f(t)=3t2+4t−1+tf(t)=3 t^{2}+4 t^{-1}+\sqrt{t}f(t)=3t2+4t−1+t​What is the object's velocity at 4 seconds?

Problem 2412

The acceleration of an object at ttt seconds is given in feet per second per second by f(t)=(t+2)(t+3)f(t)=(t+2)(t+3)f(t)=(t+2)(t+3)At 1 second the object's velocity is 9 feet per second. What is the object's velocity at 3 seconds?

Problem 2413

Identify the transformation from the graph of f(x)=x2f(x)=x^{2}f(x)=x2 to the graph of g(x)g(x)g(x). Then graph⁡f(x)\operatorname{graph} f(x)graphf(x) and g(x)g(x)g(x) on the same coordinate plane.17. g(x)=x2−7g(x)=x^{2}-7g(x)=x2−718. g(x)=x2+10g(x)=x^{2}+10g(x)=x2+10

Problem 2414

Problem 2415

3. A function f(x)f(x)f(x) is defined as f(x)=3x−10f(x)=3 x-10f(x)=3x−10. a) Given that the range of f(x)f(x)f(x) is 5<f(x)<505<f(x)<505<f(x)<50, find the domain of f(x)f(x)f(x). (2 marks) b) Find f(f(10))f(f(10))f(f(10)). (2 marks)

Problem 2416

f(t)={t2−5t,t≤−10t+19,,−10<t<−2t3t+9,t≥−2\begin{array}{l}f(t)=\left\{\begin{array}{ll}t^{2}-5 t & , \quad t \leq-10 \\ t+19 & ,\end{array},-10<t<-2\right. \\ \frac{t^{3}}{t+9}\end{array}, \quad t \geq-2f(t)={t2−5tt+19​,t≤−10,​,−10<t<−2t+9t3​​,t≥−2.

Problem 2417

Problem 2418

Problem 2419

Problem 2420

Question 6 Here is the graph of an exponential function fff. Find an equation defining fff. f(x)=f(x)=f(x)= □\square□ type your answer...

Problem 2421

Question 26 (Mandatory) (1 point) The function h(t)=−5t2+7t+12h(t)=-5 t^{2}+7 t+12h(t)=−5t2+7t+12, where h(t)h(t)h(t) is the height in metres and ttt is the time in seconds, models the height of a ball thrown from a balcony to the ground below. What is the maximum height reached by the ball?

Problem 2422

Where is the graph of f(x)=4⌊x−3⌋+2f(x)=4\lfloor x-3\rfloor+2f(x)=4⌊x−3⌋+2 discontinuous? all real numbers all integers only at multiples of 3 only at multiples of 4

Problem 2423

Problem 2424

Problem 2425

f(x)={−10+x2 if −4≤x<412−x if x≥4f(x)=\left\{\begin{array}{rrc} -10+x^{2} & \text { if } & -4 \leq x<4 \\ 12-x & \text { if } & x \geq 4 \end{array}\right.f(x)={−10+x212−x​ if if ​−4≤x<4x≥4​Graph the function fff. Then determine whether or

Problem 2426

Jimmy receives his first 3 video games free then pays $50\$ 50$50 for each game after that.Is the amount of money he spends on video games proportional to the number of games he owns?Choose 1 answer: (A) Yes (B) No

Problem 2427

nt/3354527/25279753/66e087c9cbf4583acf59eeb0809cf419The graph of y=f(x)y=f(x)y=f(x) is shown below. What are all of the real solutions of f(x)=0f(x)=0f(x)=0 ?

Problem 2428

Problem 2429

Problem 2430

Problem 2431

Problem 2432

Find the function λ\lambdaλ for the Heisenberg group representation where V=VλV=V_{\lambda}V=Vλ​ and λ\lambdaλ is not linear.

Problem 2433

Compare x−yx - yx−y and (x+y)2(x + y)^{2}(x+y)2 given x>0>yx > 0 > yx>0>y.

Problem 2434

Find the domain and range of the function f(x)=4(2−2x+7)−5f(x)=4(2^{-2x+7})-5f(x)=4(2−2x+7)−5.

Problem 2435

Identify the transformation of the function f(x)=−3(52x−6)+3f(x)=-3\left(5^{2 x-6}\right)+3f(x)=−3(52x−6)+3: a) Right 3 units b) Down 6 units c) Stretch by 3 d) Reflection on yyy-axis

Problem 2436

Find the derivative f′(x)f^{\prime}(x)f′(x) of f(x)=∣x∣f(x)=|x|f(x)=∣x∣ for x>0x>0x>0, x<0x<0x<0, and at x=0x=0x=0. Explain why it's undefined at 000.

Problem 2437

In a game, Andy scores 10% more than Bill, and Bill scores 10% less than Chris, who scored 700. Are Andy and Chris's scores equal? By what percent is the highest score greater than the lowest?

Problem 2438

Given values of f′′(x)f^{\prime \prime}(x)f′′(x): at x=−1x=-1x=−1 is -4, x=0x=0x=0 is -1, x=1x=1x=1 is 2, x=2x=2x=2 is 5, x=3x=3x=3 is 8. What type of function is fff? (A) linear (B) quadratic (C) cubic (D) fourth-degree (E) exponential

Problem 2439

Find the value of ccc such that f(x)=x+cxf(x)=x+\frac{c}{x}f(x)=x+xc​ has a local minimum at x=3x=3x=3.

Problem 2440

Find the time ttt (from 0 to 10) when the object with velocity v(t)=tcos⁡t−ln⁡(t+2)v(t)=t \cos t-\ln (t+2)v(t)=tcost−ln(t+2) reaches maximum speed.

Problem 2441

Calculate the area under the curve y=41+x2y=\frac{4}{1+x^{2}}y=1+x24​ from x=−1x=-1x=−1 to x=1x=1x=1.

Problem 2442

Find the time ttt in [0,10][0, 10][0,10] when the velocity v(t)=tcos⁡t−ln⁡(t+2)v(t)=t \cos t - \ln(t+2)v(t)=tcost−ln(t+2) is maximized. Options: A. 9.5 B. 5.1 C. 6.4 D. 7.6

Problem 2443

In a game, Peter scored xxx, Mary scored yyy, and Susan scored zzz. Show zzz in terms of xxx and yyy.

Problem 2444

In 2000, Mexico's population was 100 million, growing at 1.53%1.53 \%1.53% per year. When will it reach 109 million? Round to nearest year. a. 2006 b. 2008 c. 2005 d. 2007 (Select One)

Problem 2445

Problem 2446

Find the rate of change of sales S(t)=10,000+2000t−200t2S(t)=10,000+2000 t-200 t^{2}S(t)=10,000+2000t−200t2 and interpret S′(t)S^{\prime}(t)S′(t).

Problem 2447

Problem 2448

Find the value of the constant ccc in f(x)=2x3+3x2+cx+8f(x)=2x^3+3x^2+cx+8f(x)=2x3+3x2+cx+8 given fff intersects the xxx-axis at (−4,0)(-4,0)(−4,0), (12,0)(\frac{1}{2},0)(21​,0), and (p,0)(p,0)(p,0).

Problem 2449

Find the derivative of the function f(x)=6⋅x⋅(x−5)f(x)=6 \cdot \sqrt{x} \cdot(x-5)f(x)=6⋅x​⋅(x−5). What is f′(x)f^{\prime}(x)f′(x)?

If $f(x)=\sin \left(4 x+\frac{\pi}{2}\right)$ , find $f^{\prime}\left(\frac{\pi}{4}\right)$ . 0 4 $-4$ 1

Level J CC. 1 Evaluate a linear function LNV Analytios You have prizes t
Use the function $y=14 t$ to find the value of $y$ when $t=8$ . Video $y=$ $\square$ Submit

State the range of the following functions: a) $f(x)=\arccos x$ b) $f(x)=\sin x$ c) $f(x)=\tan ^{-1} x$

The position of an object at t seconds is given in feet by $f(t)=3 t^{2}+4 t^{-1}+\sqrt{t}$
What is the object's velocity at 4 seconds?

The acceleration of an object at $t$ seconds is given in feet per second per second by $f(t)=(t+2)(t+3)$
At 1 second the object's velocity is 9 feet per second. What is the object's velocity at 3 seconds?

Identify the transformation from the graph of $f(x)=x^{2}$ to the graph of $g(x)$ . Then $\operatorname{graph} f(x)$ and $g(x)$ on the same coordinate plane.
17. $g(x)=x^{2}-7$
18. $g(x)=x^{2}+10$

3. A function $f(x)$ is defined as $f(x)=3 x-10$ . a) Given that the range of $f(x)$ is $5<f(x)<50$ , find the domain of $f(x)$ . (2 marks) b) Find $f(f(10))$ . (2 marks)

$\begin{array}{l}f(t)=\left\{\begin{array}{ll}t^{2}-5 t & , \quad t \leq-10 \\ t+19 & ,\end{array},-10<t<-2\right. \\ \frac{t^{3}}{t+9}\end{array}, \quad t \geq-2$ .

Question 6 Here is the graph of an exponential function $f$ . Find an equation defining $f$ . $f(x)=$ $\square$ type your answer...

Question 26 (Mandatory) (1 point) The function $h(t)=-5 t^{2}+7 t+12$ , where $h(t)$ is the height in metres and $t$ is the time in seconds, models the height of a ball thrown from a balcony to the ground below. What is the maximum height reached by the ball?

Where is the graph of $f(x)=4\lfloor x-3\rfloor+2$ discontinuous? all real numbers all integers only at multiples of 3 only at multiples of 4

$f(x)=\left\{\begin{array}{rrc} -10+x^{2} & \text { if } & -4 \leq x<4 \\ 12-x & \text { if } & x \geq 4 \end{array}\right.$
Graph the function $f$ . Then determine whether or

Jimmy receives his first 3 video games free then pays $\$ 50$ for each game after that.
Is the amount of money he spends on video games proportional to the number of games he owns?
Choose 1 answer: (A) Yes (B) No

nt/3354527/25279753/66e087c9cbf4583acf59eeb0809cf419
The graph of $y=f(x)$ is shown below. What are all of the real solutions of $f(x)=0$ ?

Find the function $\lambda$ for the Heisenberg group representation where $V=V_{\lambda}$ and $\lambda$ is not linear.

Compare $x - y$ and $(x + y)^{2}$ given $x > 0 > y$ .

Find the domain and range of the function $f(x)=4(2^{-2x+7})-5$ .

Identify the transformation of the function $f(x)=-3\left(5^{2 x-6}\right)+3$ : a) Right 3 units b) Down 6 units c) Stretch by 3 d) Reflection on $y$ -axis

Find the derivative $f^{\prime}(x)$ of $f(x)=|x|$ for $x>0$ , $x<0$ , and at $x=0$ . Explain why it's undefined at $0$ .

Given values of $f^{\prime \prime}(x)$ : at $x=-1$ is -4, $x=0$ is -1, $x=1$ is 2, $x=2$ is 5, $x=3$ is 8. What type of function is $f$ ? (A) linear (B) quadratic (C) cubic (D) fourth-degree (E) exponential

Find the value of $c$ such that $f(x)=x+\frac{c}{x}$ has a local minimum at $x=3$ .

Find the time $t$ (from 0 to 10) when the object with velocity $v(t)=t \cos t-\ln (t+2)$ reaches maximum speed.

Calculate the area under the curve $y=\frac{4}{1+x^{2}}$ from $x=-1$ to $x=1$ .

Find the time $t$ in $[0, 10]$ when the velocity $v(t)=t \cos t - \ln(t+2)$ is maximized. Options: A. 9.5 B. 5.1 C. 6.4 D. 7.6

In a game, Peter scored $x$ , Mary scored $y$ , and Susan scored $z$ . Show $z$ in terms of $x$ and $y$ .

In 2000, Mexico's population was 100 million, growing at $1.53 \%$ per year. When will it reach 109 million? Round to nearest year. a. 2006 b. 2008 c. 2005 d. 2007 (Select One)

Find the rate of change of sales $S(t)=10,000+2000 t-200 t^{2}$ and interpret $S^{\prime}(t)$ .

Find the value of the constant $c$ in $f(x)=2x^3+3x^2+cx+8$ given $f$ intersects the $x$ -axis at $(-4,0)$ , $(\frac{1}{2},0)$ , and $(p,0)$ .

Find the derivative of the function $f(x)=6 \cdot \sqrt{x} \cdot(x-5)$ . What is $f^{\prime}(x)$ ?

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-3 \cdot \ln (4 \cdot x)$ and calculate $f^{\prime}(2)$ .

Find the derivative of $f(x)=\frac{\sqrt{x}-7}{\sqrt{x}+7}$ and calculate $f^{\prime}(5)$ .

Find the derivative $f'(x)$ of the function $f(x)=\sqrt{4x^2 + 4x + 4}$ and evaluate it at $x=4$ .

Calculate the derivative of $g(x)=(4x^{2}+5x) \cdot e^{x}$ .

Find values of $x$ where $f(x)=\begin{cases}x^{2}-1, & x<1 \\ x, & x \geq 1\end{cases}$ is discontinuous and explain why.

Find discontinuities of $f(x)=\left\{\begin{array}{ll}x^{2}-1, & x<1 \\ x, & x \geq 1\end{array}\right.$ and explain why.

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2+\frac{7}{x}+\frac{7}{x^{2}}$ .

Find the derivative of $R(j)=\left(\ln \left(j^{2}\right)\right)^{2}$ at $j=e$ .

Find the derivative $f^{\prime}(t)$ of the function $f(t)=(t^{2}+5t+3)(2t^{-2}+6t^{-5})$ .

Find the marginal cost for both businesses given their cost functions: $C_A(x)=200+25x+0.1x^2$ and $C_B(x)=400+80x+0.06x^2$ . For $x=500$ , which business has the lowest cost for the next tire?

Find the critical number of the function $f(x)=(8 \cdot x-7) \cdot e^{5 \cdot x}$ .

Find the length $L$ and width $W$ (with $W \leq L$ ) of a rectangle with perimeter 28 that maximizes area. What is the max area?

The demand function is $D(x)=967-25-x$ .
(a) Find the elasticity of demand. (b) Find the price where elasticity equals 1. (c) Determine the price for maximum revenue.

Projectile height is given by $h(t)=-16 \cdot t^{2}+256-t$ . Find: (a) average velocity for 3s, (b) speed & height at 6s, (c) max height & time, (d) acceleration at $t=5$ .

Approximate the integral $\int_{-1}^{2}(x^{4}-3 \cdot x^{2}+1) d x$ using left Riemann sums with 8 subdivisions. Round to three decimal places.

Find the total displacement of a ball with velocity $v(t)=90-32 \cdot t$ from $t=2$ to $t=5$ seconds.

Calculate the area between $y=\sqrt{x}$ and $y=0$ over the interval $[-1,1]$ .

Calculate the 5-unit moving average of the function $f(x) = x^3 - x$ .

Find the average rate of change of profit $P(x)=-400x^2+6800x-12000$ over $[6,6+h]$ for $h=1, 0.1, 0.01, 0.001, 0.0001$ . What do the results indicate?

A particle's position is given by $s=2 \cdot t^{3}-7 \cdot t^{2}-9 \cdot t+12$ . Find velocity, acceleration, and specific values.

Projectile height is given by $h(t)=-16 \cdot t^{2}+256 \cdot t$ . Find average velocity, speed, max height, and acceleration at $t=5$ .

Show that the limit $\operatorname{limit}_{x \rightarrow 2} \frac{|x-2|}{x-2}$ does not exist.

Find the speed of a satellite in a circular orbit just above Earth, given radius $6.4 \times 10^6 \mathrm{~m}$ and $g=9.8 \mathrm{~m/s}^2$ .

Find $\sqrt[3]{y}$ given that $(3,8)=(3, \sqrt{y})$ .

If $(x-1,11)=(8, y+3)$ , find $\sqrt{x+2y}$ . Options: (a) 5 (b) $\pm 5$ (c) $\sqrt{17}$ (d) 25