Solved on Jan 22, 2024

Find the domain, limits, and graph of the piecewise function $f(x) = \sqrt{25 - x^2}$ for $0 \leq x < 5$ , $f(x) = 5$ for $5 \leq x < 10$ , and $f(x) = 10$ for $x = 10$ .

STEP 1

Assumptions
1. The function $f(x)$ is a piecewise function defined by three different expressions over different intervals.
2. The function is defined for $0 \leq x \leq 10$ .
3. The domain of a function is the set of all possible input values (x-values) for which the function is defined.
4. The range of a function is the set of all possible output values (f(x)-values) the function can produce.

STEP 2

To find the domain of $f(x)$ , we look at the intervals where the function is defined.
The domain of $f(x)$ is the union of the intervals $[0, 5)$ , $[5, 10)$ , and $\{10\}$ .

STEP 3

Write the domain of $f(x)$ in interval notation.
$Domain\, of\, f(x) = [0, 10]$

STEP 4

To find the range of $f(x)$ , we evaluate the output values for each piece of the function.
For $0 \leq x < 5$ , the output is $\sqrt{25-x^2}$ , which varies between $0$ and $5$ .
For $5 \leq x < 10$ , the output is constant at $5$ .
For $x = 10$ , the output is $10$ .

STEP 5

Combine the output values to find the range of $f(x)$ .
The range of $f(x)$ is the union of the values from $0$ to $5$ , including $5$ , and the single value $10$ .

STEP 6

Write the range of $f(x)$ in interval notation.
$Range\, of\, f(x) = [0, 5] \cup \{10\}$

STEP 7

To determine the points $c$ where $\lim _{x \rightarrow c} f(x)$ exists, we need to check the continuity of the function at each interval and at the endpoints of the intervals.
For $0 \leq x < 5$ , the function is continuous because it is a square root function over its domain.
For $5 \leq x < 10$ , the function is constant and therefore continuous.
At $x = 10$ , the function is defined as a single value, so the limit as $x$ approaches $10$ from the left exists and is equal to $5$ .

STEP 8

Identify the points $c$ where the limit exists.
The limit exists for all $c$ in the interval $[0, 10)$ and at $c = 10$ from the left-hand side.

STEP 9

To find points where the left-hand limit exists but not the right-hand limit, we look for discontinuities where the function jumps from one value to another as $x$ increases.
At $x = 5$ , the function jumps from $\sqrt{25-5^2} = 0$ to $5$ . The left-hand limit as $x$ approaches $5$ exists and is $0$ , but the right-hand limit is $5$ .

STEP 10

Identify the point where the left-hand limit exists but not the right-hand limit.
The point is $x = 5$ .

STEP 11

To find points where the right-hand limit exists but not the left-hand limit, we look for discontinuities where the function jumps from one value to another as $x$ decreases.
Since there are no such points in the given function, we conclude that there are no points where the right-hand limit exists but not the left-hand limit.

STEP 12

Choose the correct graph of $f(x)$ based on the information obtained.
The graph should show a semicircle for $0 \leq x < 5$ , a horizontal line at $y = 5$ for $5 \leq x < 10$ , and a single point at $(10, 10)$ .
Without the actual graphs labeled A, B, C, and D, we cannot visually identify the correct graph. However, based on the description, the correct graph would match the behavior described in the previous steps.

Was this helpful?

Find the domain, limits, and graph of the piecewise function $f(x) = \sqrt{25 - x^2}$ for $0 \leq x < 5$ , $f(x) = 5$ for $5 \leq x < 10$ , and $f(x) = 10$ for $x = 10$ .

STEP 1

STEP 2

To find the domain of $f(x)$ , we look at the intervals where the function is defined.
The domain of $f(x)$ is the union of the intervals $[0, 5)$ , $[5, 10)$ , and $\{10\}$ .

STEP 3

Write the domain of $f(x)$ in interval notation.
$Domain\, of\, f(x) = [0, 10]$

STEP 4

To find the range of $f(x)$ , we evaluate the output values for each piece of the function.
For $0 \leq x < 5$ , the output is $\sqrt{25-x^2}$ , which varies between $0$ and $5$ .
For $5 \leq x < 10$ , the output is constant at $5$ .
For $x = 10$ , the output is $10$ .

STEP 5

Combine the output values to find the range of $f(x)$ .
The range of $f(x)$ is the union of the values from $0$ to $5$ , including $5$ , and the single value $10$ .

STEP 6

Write the range of $f(x)$ in interval notation.
$Range\, of\, f(x) = [0, 5] \cup \{10\}$

STEP 7

STEP 8

Identify the points $c$ where the limit exists.
The limit exists for all $c$ in the interval $[0, 10)$ and at $c = 10$ from the left-hand side.

STEP 9

STEP 10

Identify the point where the left-hand limit exists but not the right-hand limit.
The point is $x = 5$ .

STEP 11

STEP 12

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the domain, limits, and graph of the piecewise function f(x)=25−x2f(x) = \sqrt{25 - x^2}f(x)=25−x2​ for 0≤x<50 \leq x < 50≤x<5, f(x)=5f(x) = 5f(x)=5 for 5≤x<105 \leq x < 105≤x<10, and f(x)=10f(x) = 10f(x)=10 for x=10x = 10x=10.

STEP 1

STEP 2

To find the domain of f(x)f(x)f(x), we look at the intervals where the function is defined.The domain of f(x)f(x)f(x) is the union of the intervals [0,5)[0, 5)[0,5), [5,10)[5, 10)[5,10), and {10}\{10\}{10}.

STEP 3

Write the domain of f(x)f(x)f(x) in interval notation.Domain of f(x)=[0,10]Domain\, of\, f(x) = [0, 10]Domainoff(x)=[0,10]

STEP 4

STEP 5

Combine the output values to find the range of f(x)f(x)f(x).The range of f(x)f(x)f(x) is the union of the values from 000 to 555, including 555, and the single value 101010.

STEP 6

Write the range of f(x)f(x)f(x) in interval notation.Range of f(x)=[0,5]∪{10}Range\, of\, f(x) = [0, 5] \cup \{10\}Rangeoff(x)=[0,5]∪{10}

STEP 7

STEP 8

Identify the points ccc where the limit exists.The limit exists for all ccc in the interval [0,10)[0, 10)[0,10) and at c=10c = 10c=10 from the left-hand side.

STEP 9

STEP 10

Identify the point where the left-hand limit exists but not the right-hand limit.The point is x=5x = 5x=5.

STEP 11

STEP 12

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the domain, limits, and graph of the piecewise function $f(x) = \sqrt{25 - x^2}$ for $0 \leq x < 5$ , $f(x) = 5$ for $5 \leq x < 10$ , and $f(x) = 10$ for $x = 10$ .

To find the domain of $f(x)$ , we look at the intervals where the function is defined.
The domain of $f(x)$ is the union of the intervals $[0, 5)$ , $[5, 10)$ , and $\{10\}$ .

Write the domain of $f(x)$ in interval notation.
$Domain\, of\, f(x) = [0, 10]$

Combine the output values to find the range of $f(x)$ .
The range of $f(x)$ is the union of the values from $0$ to $5$ , including $5$ , and the single value $10$ .

Write the range of $f(x)$ in interval notation.
$Range\, of\, f(x) = [0, 5] \cup \{10\}$

Identify the points $c$ where the limit exists.
The limit exists for all $c$ in the interval $[0, 10)$ and at $c = 10$ from the left-hand side.

Identify the point where the left-hand limit exists but not the right-hand limit.
The point is $x = 5$ .