Question

4.
$f(x) = \

$$\begin{cases} |2x + 7|, & x \\leq -4 \\\\ 1 + x^2, & -4 < x \\leq 1 \\\\ 6, & 1 < x < 3 \\\\ \\dfrac{1}{3}x + 8, & x \\geq 3 \\end{cases}$$

$

a. $f(5) = $

b. $f(1) = $

c. $f(-4) = $

d. $f(2) = $

Explanation:

Step1: Match x=5 to correct piece

Since $5 \geq 3$, use $f(x)=\frac{1}{3}x + 8$.

Step2: Calculate f(5)

$
f(5)=\frac{1}{3}(5) + 8 = \frac{5}{3} + 8 = \frac{5}{3} + \frac{24}{3} = \frac{29}{3}
$

Step3: Match x=1 to correct piece

Since $-4 < 1 \leq 1$, use $f(x)=1+x^2$.

Step4: Calculate f(1)

$
f(1)=1+(1)^2 = 1+1=2
$

Step5: Match x=-4 to correct piece

Since $-4 \leq -4$, use $f(x)=|2x+7|$.

Step6: Calculate f(-4)

$
f(-4)=|2(-4)+7| = |-8+7|=|-1|=1
$

Step7: Match x=2 to correct piece

Since $1 < 2 < 3$, use $f(x)=6$.

Step8: Calculate f(2)

$
f(2)=6
$

Answer:

a. $\frac{29}{3}$
b. $2$
c. $1$
d. $6$