QuestionFind $f(0)$ , $f(2)$ , and $f(-2)$ for the piecewise function: $f(x)=\left\{\begin{array}{lr} -4, & x \leq-2 \\ x-2, & -2<x<2 \\ -2 x+4, & x \geq 2 \end{array}\right.$

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as a piecewise function with three different expressions for different intervals of $x$ . . We need to find the values of $f(0)$ , $f()$ , and $f(-)$ .

STEP 2

To find $f(0)$ , we need to determine which interval $0$ falls in. Looking at the intervals in the piecewise function, we see that $0$ falls in the interval $-2<x<2$ .

STEP 3

Substitute $0$ into the expression for the interval $-2<x<2$ , which is $x-2$ .
$f(0) =0 -2$

STEP 4

Calculate the value of $f(0)$ .
$f(0) =0 -2 = -2$

STEP 5

To find $f(2)$ , we need to determine which interval $2$ falls in. Looking at the intervals in the piecewise function, we see that $2$ falls in the interval $x \geq2$ .

STEP 6

Substitute $2$ into the expression for the interval $x \geq2$ , which is $-2x+4$ .
$f(2) = -2(2) +4$

STEP 7

Calculate the value of $f(2)$ .
$f(2) = -2(2) +4 =0$

STEP 8

To find $f(-2)$ , we need to determine which interval $-2$ falls in. Looking at the intervals in the piecewise function, we see that $-2$ falls in the interval $x \leq -2$ .

STEP 9

Substitute $-2$ into the expression for the interval $x \leq -2$ , which is $-4$ .
$f(-2) = -4$

STEP 10

Since the expression for the interval $x \leq -2$ is a constant, the value of $f(-2)$ is simply $-4$ .
So, $f(0) = -2$ , $f(2) =0$ , and $f(-2) = -4$ .

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QuestionFind $f(0)$ , $f(2)$ , and $f(-2)$ for the piecewise function: $f(x)=\left\{\begin{array}{lr} -4, & x \leq-2 \\ x-2, & -2<x<2 \\ -2 x+4, & x \geq 2 \end{array}\right.$

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as a piecewise function with three different expressions for different intervals of $x$ . . We need to find the values of $f(0)$ , $f()$ , and $f(-)$ .

STEP 2

To find $f(0)$ , we need to determine which interval $0$ falls in. Looking at the intervals in the piecewise function, we see that $0$ falls in the interval $-2<x<2$ .

STEP 3

Substitute $0$ into the expression for the interval $-2<x<2$ , which is $x-2$ .
$f(0) =0 -2$

STEP 4

Calculate the value of $f(0)$ .
$f(0) =0 -2 = -2$

STEP 5

To find $f(2)$ , we need to determine which interval $2$ falls in. Looking at the intervals in the piecewise function, we see that $2$ falls in the interval $x \geq2$ .

STEP 6

Substitute $2$ into the expression for the interval $x \geq2$ , which is $-2x+4$ .
$f(2) = -2(2) +4$

STEP 7

Calculate the value of $f(2)$ .
$f(2) = -2(2) +4 =0$

STEP 8

To find $f(-2)$ , we need to determine which interval $-2$ falls in. Looking at the intervals in the piecewise function, we see that $-2$ falls in the interval $x \leq -2$ .

STEP 9

Substitute $-2$ into the expression for the interval $x \leq -2$ , which is $-4$ .
$f(-2) = -4$

STEP 10

Since the expression for the interval $x \leq -2$ is a constant, the value of $f(-2)$ is simply $-4$ .
So, $f(0) = -2$ , $f(2) =0$ , and $f(-2) = -4$ .

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Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x)f(x) is defined as a piecewise function with three different expressions for different intervals of xxx. . We need to find the values of f(0)f(0)f(0), f()f()f(), and f(−)f(-)f(−).

STEP 2

To find f(0)f(0)f(0), we need to determine which interval 000 falls in. Looking at the intervals in the piecewise function, we see that 000 falls in the interval −2<x<2-2<x<2−2<x<2.

STEP 3

Substitute 000 into the expression for the interval −2<x<2-2<x<2−2<x<2, which is x−2x-2x−2.f(0)=0−2f(0) =0 -2f(0)=0−2

STEP 4

Calculate the value of f(0)f(0)f(0).f(0)=0−2=−2f(0) =0 -2 = -2f(0)=0−2=−2

STEP 5

To find f(2)f(2)f(2), we need to determine which interval 222 falls in. Looking at the intervals in the piecewise function, we see that 222 falls in the interval x≥2x \geq2x≥2.

STEP 6

Substitute 222 into the expression for the interval x≥2x \geq2x≥2, which is −2x+4-2x+4−2x+4.f(2)=−2(2)+4f(2) = -2(2) +4f(2)=−2(2)+4

STEP 7

Calculate the value of f(2)f(2)f(2).f(2)=−2(2)+4=0f(2) = -2(2) +4 =0f(2)=−2(2)+4=0

STEP 8

To find f(−2)f(-2)f(−2), we need to determine which interval −2-2−2 falls in. Looking at the intervals in the piecewise function, we see that −2-2−2 falls in the interval x≤−2x \leq -2x≤−2.

STEP 9

Substitute −2-2−2 into the expression for the interval x≤−2x \leq -2x≤−2, which is −4-4−4.f(−2)=−4f(-2) = -4f(−2)=−4

STEP 10

Since the expression for the interval x≤−2x \leq -2x≤−2 is a constant, the value of f(−2)f(-2)f(−2) is simply −4-4−4.So, f(0)=−2f(0) = -2f(0)=−2, f(2)=0f(2) =0f(2)=0, and f(−2)=−4f(-2) = -4f(−2)=−4.

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Assumptions1. The function $f(x)$ is defined as a piecewise function with three different expressions for different intervals of $x$ . . We need to find the values of $f(0)$ , $f()$ , and $f(-)$ .

To find $f(0)$ , we need to determine which interval $0$ falls in. Looking at the intervals in the piecewise function, we see that $0$ falls in the interval $-2<x<2$ .

Substitute $0$ into the expression for the interval $-2<x<2$ , which is $x-2$ .
$f(0) =0 -2$

Calculate the value of $f(0)$ .
$f(0) =0 -2 = -2$

To find $f(2)$ , we need to determine which interval $2$ falls in. Looking at the intervals in the piecewise function, we see that $2$ falls in the interval $x \geq2$ .

Substitute $2$ into the expression for the interval $x \geq2$ , which is $-2x+4$ .
$f(2) = -2(2) +4$

Calculate the value of $f(2)$ .
$f(2) = -2(2) +4 =0$

To find $f(-2)$ , we need to determine which interval $-2$ falls in. Looking at the intervals in the piecewise function, we see that $-2$ falls in the interval $x \leq -2$ .

Substitute $-2$ into the expression for the interval $x \leq -2$ , which is $-4$ .
$f(-2) = -4$

Since the expression for the interval $x \leq -2$ is a constant, the value of $f(-2)$ is simply $-4$ .
So, $f(0) = -2$ , $f(2) =0$ , and $f(-2) = -4$ .