QuestionFind the domain of $f(x)$ and evaluate $f(-2)$ and $f(0)$ for the piecewise function defined as: $f(x)=\left\{\begin{array}{rr} 2 x & \text{if } -2 \leq x \leq 3 \\ 4 & \text{if } 3<x \leq 5 \end{array}\right.$

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $f(x)=\left\{\begin{array}{rr} x \text { if } & - \leq x \leq3 \\ 4 \text { if } &3<x \leq5\end{array}\right.$

STEP 2

To determine the domain of $f(x)$ , we need to look at the range of $x$ values for which the function is defined. The function is defined for $-2 \leq x \leq$ and $< x \leq5$ .

STEP 3

Combine the two intervals to get the domain of $f(x)$ .
$Domain = [-2,3] \cup (3,5]$

STEP 4

To evaluate $f(-2)$ , we need to find the value of $f(x)$ when $x = -2$ . Since $-2$ falls in the interval $-2 \leq x \leq3$ , we use the first part of the function definition.
$f(-2) =2 \times -2$

STEP 5

Calculate the value of $f(-2)$ .
$f(-2) =2 \times -2 = -4$

STEP 6

To evaluate $f(0)$ , we need to find the value of $f(x)$ when $x =0$ . Since $0$ falls in the interval $-2 \leq x \leq3$ , we use the first part of the function definition.
$f(0) =2 \times0$

STEP 7

Calculate the value of $f(0)$ .
$f(0) =2 \times0 =0$ Solution(a) The domain of $f$ is $[-2,3] \cup (3,5]$ . (b) $f(-2) = -4$ . (c) $f(0) =0$ .

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QuestionFind the domain of $f(x)$ and evaluate $f(-2)$ and $f(0)$ for the piecewise function defined as: $f(x)=\left\{\begin{array}{rr} 2 x & \text{if } -2 \leq x \leq 3 \\ 4 & \text{if } 3<x \leq 5 \end{array}\right.$

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $f(x)=\left\{\begin{array}{rr} x \text { if } & - \leq x \leq3 \\ 4 \text { if } &3<x \leq5\end{array}\right.$

STEP 2

To determine the domain of $f(x)$ , we need to look at the range of $x$ values for which the function is defined. The function is defined for $-2 \leq x \leq$ and $< x \leq5$ .

STEP 3

Combine the two intervals to get the domain of $f(x)$ .
$Domain = [-2,3] \cup (3,5]$

STEP 4

To evaluate $f(-2)$ , we need to find the value of $f(x)$ when $x = -2$ . Since $-2$ falls in the interval $-2 \leq x \leq3$ , we use the first part of the function definition.
$f(-2) =2 \times -2$

STEP 5

Calculate the value of $f(-2)$ .
$f(-2) =2 \times -2 = -4$

STEP 6

To evaluate $f(0)$ , we need to find the value of $f(x)$ when $x =0$ . Since $0$ falls in the interval $-2 \leq x \leq3$ , we use the first part of the function definition.
$f(0) =2 \times0$

STEP 7

Calculate the value of $f(0)$ .
$f(0) =2 \times0 =0$ Solution(a) The domain of $f$ is $[-2,3] \cup (3,5]$ . (b) $f(-2) = -4$ . (c) $f(0) =0$ .

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Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x)f(x) is defined asf(x)={x if −≤x≤34 if 3<x≤5f(x)=\left\{\begin{array}{rr} x \text { if } & - \leq x \leq3 \\ 4 \text { if } &3<x \leq5\end{array}\right. f(x)={x if 4 if ​−≤x≤33<x≤5​

STEP 2

To determine the domain of f(x)f(x)f(x), we need to look at the range of xxx values for which the function is defined. The function is defined for −2≤x≤-2 \leq x \leq−2≤x≤ and <x≤5 < x \leq5<x≤5.

STEP 3

Combine the two intervals to get the domain of f(x)f(x)f(x).Domain=[−2,3]∪(3,5]Domain = [-2,3] \cup (3,5]Domain=[−2,3]∪(3,5]

STEP 4

To evaluate f(−2)f(-2)f(−2), we need to find the value of f(x)f(x)f(x) when x=−2x = -2x=−2. Since −2-2−2 falls in the interval −2≤x≤3-2 \leq x \leq3−2≤x≤3, we use the first part of the function definition.f(−2)=2×−2f(-2) =2 \times -2f(−2)=2×−2

STEP 5

Calculate the value of f(−2)f(-2)f(−2).f(−2)=2×−2=−4f(-2) =2 \times -2 = -4f(−2)=2×−2=−4

STEP 6

To evaluate f(0)f(0)f(0), we need to find the value of f(x)f(x)f(x) when x=0x =0x=0. Since 000 falls in the interval −2≤x≤3-2 \leq x \leq3−2≤x≤3, we use the first part of the function definition.f(0)=2×0f(0) =2 \times0f(0)=2×0

STEP 7

Calculate the value of f(0)f(0)f(0).f(0)=2×0=0f(0) =2 \times0 =0f(0)=2×0=0Solution(a) The domain of fff is [−2,3]∪(3,5][-2,3] \cup (3,5][−2,3]∪(3,5]. (b) f(−2)=−4f(-2) = -4f(−2)=−4. (c) f(0)=0f(0) =0f(0)=0.

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Assumptions1. The function $f(x)$ is defined as $f(x)=\left\{\begin{array}{rr} x \text { if } & - \leq x \leq3 \\ 4 \text { if } &3<x \leq5\end{array}\right.$

To determine the domain of $f(x)$ , we need to look at the range of $x$ values for which the function is defined. The function is defined for $-2 \leq x \leq$ and $< x \leq5$ .

Combine the two intervals to get the domain of $f(x)$ .
$Domain = [-2,3] \cup (3,5]$

To evaluate $f(-2)$ , we need to find the value of $f(x)$ when $x = -2$ . Since $-2$ falls in the interval $-2 \leq x \leq3$ , we use the first part of the function definition.
$f(-2) =2 \times -2$

Calculate the value of $f(-2)$ .
$f(-2) =2 \times -2 = -4$

To evaluate $f(0)$ , we need to find the value of $f(x)$ when $x =0$ . Since $0$ falls in the interval $-2 \leq x \leq3$ , we use the first part of the function definition.
$f(0) =2 \times0$

Calculate the value of $f(0)$ .
$f(0) =2 \times0 =0$ Solution(a) The domain of $f$ is $[-2,3] \cup (3,5]$ . (b) $f(-2) = -4$ . (c) $f(0) =0$ .