Question

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Explanation:

Exercise 47: $f(x) =

$$\begin{cases} -3x + 7 & \text{for } x < -1 \\ x^2 + 3 & \text{for } -1 \leq x < 4 \\ 5 & \text{for } x \geq 4 \end{cases}$$

$

Part a: $f(3)$

Step1: Match x to correct interval

$3$ is in $-1 \leq x < 4$, use $x^2+3$.

Step2: Substitute $x=3$

$f(3) = 3^2 + 3 = 9 + 3$

Part b: $f(-2)$

Step1: Match x to correct interval

$-2$ is in $x < -1$, use $-3x+7$.

Step2: Substitute $x=-2$

$f(-2) = -3(-2) + 7 = 6 + 7$

Part c: $f(-1)$

Step1: Match x to correct interval

$-1$ is in $-1 \leq x < 4$, use $x^2+3$.

Step2: Substitute $x=-1$

$f(-1) = (-1)^2 + 3 = 1 + 3$

Part d: $f(4)$

Step1: Match x to correct interval

$4$ is in $x \geq 4$, use $5$.

Step2: Evaluate the constant

$f(4) = 5$

Part e: $f(5)$

Step1: Match x to correct interval

$5$ is in $x \geq 4$, use $5$.

Step2: Evaluate the constant

$f(5) = 5$

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Exercise 48: $g(x) =

$$\begin{cases} -2|x| - 3 & \text{for } x \leq -2 \\ 5x + 6 & \text{for } -2 < x < 3 \\ 4 & \text{for } x \geq 3 \end{cases}$$

$

Part a: $g(-3)$

Step1: Match x to correct interval

$-3$ is in $x \leq -2$, use $-2|x|-3$.

Step2: Substitute $x=-3$

$g(-3) = -2|-3| - 3 = -2(3) - 3$

Part b: $g(3)$

Step1: Match x to correct interval

$3$ is in $x \geq 3$, use $4$.

Step2: Evaluate the constant

$g(3) = 4$

Part c: $g(-2)$

Step1: Match x to correct interval

$-2$ is in $x \leq -2$, use $-2|x|-3$.

Step2: Substitute $x=-2$

$g(-2) = -2|-2| - 3 = -2(2) - 3$

Part d: $g(0)$

Step1: Match x to correct interval

$0$ is in $-2 < x < 3$, use $5x+6$.

Step2: Substitute $x=0$

$g(0) = 5(0) + 6$

Part e: $g(4)$

Step1: Match x to correct interval

$4$ is in $x \geq 3$, use $4$.

Step2: Evaluate the constant

$g(4) = 4$

Answer:

Exercise 47:
a. $f(3)=12$
b. $f(-2)=13$
c. $f(-1)=4$
d. $f(4)=5$
e. $f(5)=5$

Exercise 48:
a. $g(-3)=-9$
b. $g(3)=4$
c. $g(-2)=-7$
d. $g(0)=6$
e. $g(4)=4$