QuestionIdentify the types of functions $f(x)=x^{3}+x^{2}-3 x+4$ and $g(x)=2^{x}-4$ . What do they have in common?

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is given as $f(x)=x^{3}+x^{}-3 x+4$ . The function $g(x)$ is given as $g(x)=^{x}-4$
3. We need to identify the type of each function and justify our answer4. We also need to identify any key features that $f(x)$ and $g(x)$ have in common, considering domain, range, $x$ -intercepts, and $y$ -intercepts

STEP 2

Let's start by identifying the type of the function $f(x)$ .The function $f(x)$ is a polynomial function because it is a sum of terms, each of which is a nonnegative integer power of $x$ .

STEP 3

Now let's identify the type of the function $g(x)$ .
The function $g(x)$ is an exponential function because it involves an exponentiation where the base is a constant and the exponent is a variable.

STEP 4

Now let's find the domain of $f(x)$ .
The domain of a polynomial function is all real numbers, because you can substitute any real number for $x$ and the function will still be defined.So, the domain of $f(x)$ is $(-\infty, +\infty)$ .

STEP 5

Let's find the domain of $g(x)$ .
The domain of an exponential function is also all real numbers, because you can substitute any real number for $x$ and the function will still be defined.So, the domain of $g(x)$ is $(-\infty, +\infty)$ .

STEP 6

Now let's find the range of $f(x)$ .
The range of a cubic polynomial function (a polynomial of degree3) is all real numbers, because the graph of a cubic function extends from $-\infty$ to $+\infty$ as $x$ goes from $-\infty$ to $+\infty$ .
So, the range of $f(x)$ is $(-\infty, +\infty)$ .

STEP 7

Let's find the range of $g(x)$ .
The range of the function $g(x)$ is $(-4, +\infty)$ because the base of the exponent (2) is greater than1, so the function increases as $x$ increases, and the smallest value the function can take is $-4$ (when $x=0$ ).

STEP 8

Now let's find the $x$ -intercepts of $f(x)$ .
The $x$ -intercepts of a function are the values of $x$ for which the function equals zero. To find the $x$ -intercepts of $f(x)$ , we set $f(x)$ equal to zero and solve for $x$ .
$x^{3}+x^{2}-3 x+4=0$ This is a cubic equation, and it may have one or three real roots. The exact roots can be found using methods like factoring, synthetic division, or the rational root theorem, but these methods are beyond the scope of this problem.

STEP 9

Let's find the $x$ -intercepts of $g(x)$ .
To find the $x$ -intercepts of $g(x)$ , we set $g(x)$ equal to zero and solve for $x$ .
$2^{x}-4=$ This equation can be solved by taking the logarithm of both sides, but this is also beyond the scope of this problem.

STEP 10

Now let's find the $y$ -intercept of $f(x)$ .
The $y$ -intercept of a function is the value of the function when $x=0$ . To find the $y$ -intercept of $f(x)$ , we substitute $x=0$ into $f(x)$ .
$f(0)=0^{3}+0^{2}-3 \cdot0+4=4$ So, the $y$ -intercept of $f(x)$ is4.

STEP 11

Let's find the $y$ -intercept of $g(x)$ .
To find the $y$ -intercept of $g(x)$ , we substitute $x=0$ into $g(x)$ .
$g(0)=^{0}-4=-4=-3$ So, the $y$ -intercept of $g(x)$ is -3.

STEP 12

Now let's identify the key features that $f(x)$ and $g(x)$ have in common.
. Both functions have the same domain, which is all real numbers.
2. Neither function has any $x$ -intercepts that can be easily determined without using advanced methods. . Both functions have different $y$ -intercepts, and these $y$ -intercepts are real numbers.

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QuestionIdentify the types of functions $f(x)=x^{3}+x^{2}-3 x+4$ and $g(x)=2^{x}-4$ . What do they have in common?

Studdy Solution

STEP 1

STEP 2

Let's start by identifying the type of the function $f(x)$ .The function $f(x)$ is a polynomial function because it is a sum of terms, each of which is a nonnegative integer power of $x$ .

STEP 3

Now let's identify the type of the function $g(x)$ .
The function $g(x)$ is an exponential function because it involves an exponentiation where the base is a constant and the exponent is a variable.

STEP 4

Now let's find the domain of $f(x)$ .
The domain of a polynomial function is all real numbers, because you can substitute any real number for $x$ and the function will still be defined.So, the domain of $f(x)$ is $(-\infty, +\infty)$ .

STEP 5

Let's find the domain of $g(x)$ .
The domain of an exponential function is also all real numbers, because you can substitute any real number for $x$ and the function will still be defined.So, the domain of $g(x)$ is $(-\infty, +\infty)$ .

STEP 6

Now let's find the range of $f(x)$ .
The range of a cubic polynomial function (a polynomial of degree3) is all real numbers, because the graph of a cubic function extends from $-\infty$ to $+\infty$ as $x$ goes from $-\infty$ to $+\infty$ .
So, the range of $f(x)$ is $(-\infty, +\infty)$ .

STEP 7

Let's find the range of $g(x)$ .
The range of the function $g(x)$ is $(-4, +\infty)$ because the base of the exponent (2) is greater than1, so the function increases as $x$ increases, and the smallest value the function can take is $-4$ (when $x=0$ ).

STEP 8

STEP 9

Let's find the $x$ -intercepts of $g(x)$ .
To find the $x$ -intercepts of $g(x)$ , we set $g(x)$ equal to zero and solve for $x$ .
$2^{x}-4=$ This equation can be solved by taking the logarithm of both sides, but this is also beyond the scope of this problem.

STEP 10

Now let's find the $y$ -intercept of $f(x)$ .
The $y$ -intercept of a function is the value of the function when $x=0$ . To find the $y$ -intercept of $f(x)$ , we substitute $x=0$ into $f(x)$ .
$f(0)=0^{3}+0^{2}-3 \cdot0+4=4$ So, the $y$ -intercept of $f(x)$ is4.

STEP 11

Let's find the $y$ -intercept of $g(x)$ .
To find the $y$ -intercept of $g(x)$ , we substitute $x=0$ into $g(x)$ .
$g(0)=^{0}-4=-4=-3$ So, the $y$ -intercept of $g(x)$ is -3.

STEP 12

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QuestionIdentify the types of functions f(x)=x3+x2−3x+4f(x)=x^{3}+x^{2}-3 x+4f(x)=x3+x2−3x+4 and g(x)=2x−4g(x)=2^{x}-4g(x)=2x−4. What do they have in common?

Studdy Solution

STEP 1

STEP 2

Let's start by identifying the type of the function f(x)f(x)f(x).The function f(x)f(x)f(x) is a polynomial function because it is a sum of terms, each of which is a nonnegative integer power of xxx.

STEP 3

Now let's identify the type of the function g(x)g(x)g(x).The function g(x)g(x)g(x) is an exponential function because it involves an exponentiation where the base is a constant and the exponent is a variable.

STEP 4

Now let's find the domain of f(x)f(x)f(x).The domain of a polynomial function is all real numbers, because you can substitute any real number for xxx and the function will still be defined.So, the domain of f(x)f(x)f(x) is (−∞,+∞)(-\infty, +\infty)(−∞,+∞).

STEP 5

Let's find the domain of g(x)g(x)g(x).The domain of an exponential function is also all real numbers, because you can substitute any real number for xxx and the function will still be defined.So, the domain of g(x)g(x)g(x) is (−∞,+∞)(-\infty, +\infty)(−∞,+∞).

STEP 6

STEP 7

Let's find the range of g(x)g(x)g(x).The range of the function g(x)g(x)g(x) is (−4,+∞)(-4, +\infty)(−4,+∞) because the base of the exponent (2) is greater than1, so the function increases as xxx increases, and the smallest value the function can take is −4-4−4 (when x=0x=0x=0).

STEP 8

STEP 9

Let's find the xxx-intercepts of g(x)g(x)g(x).To find the xxx-intercepts of g(x)g(x)g(x), we set g(x)g(x)g(x) equal to zero and solve for xxx.2x−4=2^{x}-4=2x−4=This equation can be solved by taking the logarithm of both sides, but this is also beyond the scope of this problem.

STEP 10

STEP 11

Let's find the yyy-intercept of g(x)g(x)g(x).To find the yyy-intercept of g(x)g(x)g(x), we substitute x=0x=0x=0 into g(x)g(x)g(x).g(0)=0−4=−4=−3g(0)=^{0}-4=-4=-3g(0)=0−4=−4=−3So, the yyy-intercept of g(x)g(x)g(x) is -3.

STEP 12

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QuestionIdentify the types of functions $f(x)=x^{3}+x^{2}-3 x+4$ and $g(x)=2^{x}-4$ . What do they have in common?

Let's start by identifying the type of the function $f(x)$ .The function $f(x)$ is a polynomial function because it is a sum of terms, each of which is a nonnegative integer power of $x$ .

Now let's identify the type of the function $g(x)$ .
The function $g(x)$ is an exponential function because it involves an exponentiation where the base is a constant and the exponent is a variable.

Now let's find the domain of $f(x)$ .
The domain of a polynomial function is all real numbers, because you can substitute any real number for $x$ and the function will still be defined.So, the domain of $f(x)$ is $(-\infty, +\infty)$ .

Let's find the domain of $g(x)$ .
The domain of an exponential function is also all real numbers, because you can substitute any real number for $x$ and the function will still be defined.So, the domain of $g(x)$ is $(-\infty, +\infty)$ .

Let's find the range of $g(x)$ .
The range of the function $g(x)$ is $(-4, +\infty)$ because the base of the exponent (2) is greater than1, so the function increases as $x$ increases, and the smallest value the function can take is $-4$ (when $x=0$ ).

Let's find the $x$ -intercepts of $g(x)$ .
To find the $x$ -intercepts of $g(x)$ , we set $g(x)$ equal to zero and solve for $x$ .
$2^{x}-4=$ This equation can be solved by taking the logarithm of both sides, but this is also beyond the scope of this problem.

Let's find the $y$ -intercept of $g(x)$ .
To find the $y$ -intercept of $g(x)$ , we substitute $x=0$ into $g(x)$ .
$g(0)=^{0}-4=-4=-3$ So, the $y$ -intercept of $g(x)$ is -3.