Solved on Jan 18, 2024

Find $f(-3)$ for the piecewise function $f(x) = \begin{cases} 9x+4 & \text{if } x \leq -5 \\ -\frac{1}{2}x+2 & \text{if } x > -5 \end{cases}$ .

STEP 1

Assumptions
1. The function $f(x)$ is defined piecewise with two different expressions.
2. The first expression is $9x + 4$ for $x \leq -5$ .
3. The second expression is $-\frac{1}{2}x + 2$ for $x > -5$ .
4. We are asked to find the value of $f(-3)$ .

STEP 2

Determine which expression to use for $f(-3)$ by checking the condition associated with each piece of the function.

STEP 3

Since $-3$ is greater than $-5$ , we use the second expression for $f(x)$ , which is $-\frac{1}{2}x + 2$ .

STEP 4

Substitute $x = -3$ into the second expression to evaluate $f(-3)$ .
$f(-3) = -\frac{1}{2}(-3) + 2$

STEP 5

Multiply $-\frac{1}{2}$ by $-3$ .
$f(-3) = \frac{3}{2} + 2$

STEP 6

Convert the integer $2$ to a fraction with the same denominator as $\frac{3}{2}$ to add the fractions.
$2 = \frac{4}{2}$

STEP 7

Add the fractions $\frac{3}{2}$ and $\frac{4}{2}$ .
$f(-3) = \frac{3}{2} + \frac{4}{2}$

STEP 8

Calculate the sum of the fractions.
$f(-3) = \frac{3 + 4}{2}$

STEP 9

Add the numerators $3$ and $4$ .
$f(-3) = \frac{7}{2}$

STEP 10

Convert the improper fraction $\frac{7}{2}$ to a mixed number or decimal if necessary.
$f(-3) = 3.5$
So, $f(-3) = 3.5$ .

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Find $f(-3)$ for the piecewise function $f(x) = \begin{cases} 9x+4 & \text{if } x \leq -5 \\ -\frac{1}{2}x+2 & \text{if } x > -5 \end{cases}$ .

STEP 1

Assumptions
1. The function $f(x)$ is defined piecewise with two different expressions.
2. The first expression is $9x + 4$ for $x \leq -5$ .
3. The second expression is $-\frac{1}{2}x + 2$ for $x > -5$ .
4. We are asked to find the value of $f(-3)$ .

STEP 2

Determine which expression to use for $f(-3)$ by checking the condition associated with each piece of the function.

STEP 3

Since $-3$ is greater than $-5$ , we use the second expression for $f(x)$ , which is $-\frac{1}{2}x + 2$ .

STEP 4

Substitute $x = -3$ into the second expression to evaluate $f(-3)$ .
$f(-3) = -\frac{1}{2}(-3) + 2$

STEP 5

Multiply $-\frac{1}{2}$ by $-3$ .
$f(-3) = \frac{3}{2} + 2$

STEP 6

Convert the integer $2$ to a fraction with the same denominator as $\frac{3}{2}$ to add the fractions.
$2 = \frac{4}{2}$

STEP 7

Add the fractions $\frac{3}{2}$ and $\frac{4}{2}$ .
$f(-3) = \frac{3}{2} + \frac{4}{2}$

STEP 8

Calculate the sum of the fractions.
$f(-3) = \frac{3 + 4}{2}$

STEP 9

Add the numerators $3$ and $4$ .
$f(-3) = \frac{7}{2}$

STEP 10

Convert the improper fraction $\frac{7}{2}$ to a mixed number or decimal if necessary.
$f(-3) = 3.5$
So, $f(-3) = 3.5$ .

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Find f(−3)f(-3)f(−3) for the piecewise function f(x)={9x+4if x≤−5−12x+2if x>−5f(x) = \begin{cases} 9x+4 & \text{if } x \leq -5 \\ -\frac{1}{2}x+2 & \text{if } x > -5 \end{cases}f(x)={9x+4−21​x+2​if x≤−5if x>−5​.

STEP 1

STEP 2

Determine which expression to use for f(−3)f(-3)f(−3) by checking the condition associated with each piece of the function.

STEP 3

Since −3-3−3 is greater than −5-5−5, we use the second expression for f(x)f(x)f(x), which is −12x+2-\frac{1}{2}x + 2−21​x+2.

STEP 4

Substitute x=−3x = -3x=−3 into the second expression to evaluate f(−3)f(-3)f(−3).f(−3)=−12(−3)+2f(-3) = -\frac{1}{2}(-3) + 2f(−3)=−21​(−3)+2

STEP 5

Multiply −12-\frac{1}{2}−21​ by −3-3−3.f(−3)=32+2f(-3) = \frac{3}{2} + 2f(−3)=23​+2

STEP 6

Convert the integer 222 to a fraction with the same denominator as 32\frac{3}{2}23​ to add the fractions.2=422 = \frac{4}{2}2=24​

STEP 7

Add the fractions 32\frac{3}{2}23​ and 42\frac{4}{2}24​.f(−3)=32+42f(-3) = \frac{3}{2} + \frac{4}{2}f(−3)=23​+24​

STEP 8

Calculate the sum of the fractions.f(−3)=3+42f(-3) = \frac{3 + 4}{2}f(−3)=23+4​

STEP 9

Add the numerators 333 and 444.f(−3)=72f(-3) = \frac{7}{2}f(−3)=27​

STEP 10

Convert the improper fraction 72\frac{7}{2}27​ to a mixed number or decimal if necessary.f(−3)=3.5f(-3) = 3.5f(−3)=3.5So, f(−3)=3.5f(-3) = 3.5f(−3)=3.5.

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Find $f(-3)$ for the piecewise function $f(x) = \begin{cases} 9x+4 & \text{if } x \leq -5 \\ -\frac{1}{2}x+2 & \text{if } x > -5 \end{cases}$ .

Determine which expression to use for $f(-3)$ by checking the condition associated with each piece of the function.

Since $-3$ is greater than $-5$ , we use the second expression for $f(x)$ , which is $-\frac{1}{2}x + 2$ .

Substitute $x = -3$ into the second expression to evaluate $f(-3)$ .
$f(-3) = -\frac{1}{2}(-3) + 2$

Multiply $-\frac{1}{2}$ by $-3$ .
$f(-3) = \frac{3}{2} + 2$

Convert the integer $2$ to a fraction with the same denominator as $\frac{3}{2}$ to add the fractions.
$2 = \frac{4}{2}$

Add the fractions $\frac{3}{2}$ and $\frac{4}{2}$ .
$f(-3) = \frac{3}{2} + \frac{4}{2}$

Calculate the sum of the fractions.
$f(-3) = \frac{3 + 4}{2}$

Add the numerators $3$ and $4$ .
$f(-3) = \frac{7}{2}$

Convert the improper fraction $\frac{7}{2}$ to a mixed number or decimal if necessary.
$f(-3) = 3.5$
So, $f(-3) = 3.5$ .