Question

1. use the graph of function g to answer these questions.

a. complete the rules above for g(x) so that the graph represents it.

b. what are the values of
g(1) g(-12) g(15)

c. for what x - values is g(x) = - 6?

1. let functions f and g be defined by the function rules below

f(x)=\begin{cases}7, & x < 3\\4x - 9, & 3leq x < 10\\-x^{2}, & xgeq 10end{cases}

g(x)=\begin{cases}0, & xleq 5\\-3x + 16, & x > 5end{cases}

a. evaluate the following
3f(0)+g(10) f(3)-5g(-1) 0.5f(20)+g(6)

Explanation:

---

Problem 1

Step1: Match graph to piecewise rule

For $-10 \leq x < -8$, the graph has $y=-8$.
For $-6$, the interval is $-8 \leq x < -1$.
For $-1 \leq x < 1$, the graph has $y=2$.
For $y=4$, the interval is $1 \leq x < 10$.

Step2: Evaluate $g(1), g(-12), g(15)$

• $g(1)$: $x=1$ is in $1 \leq x <10$, so $g(1)=4$.
• $g(-12)$: $x=-12$ is in $-15 \leq x < -10$, so $g(-12)=-10$.
• $g(15)$: $x=15$ is not in any interval of $g(x)$, so it is undefined.

Step3: Find $x$ for $g(x)=-6$

$g(x)=-6$ corresponds to the interval $-8 \leq x < -1$.

Step1: Evaluate $3f(0)+g(10)$

• $f(0)$: $0<3$, so $f(0)=7$.
• $g(10)$: $10>5$, so $g(10)=-3(10)+16=-14$.
• Expression: $3(7) + (-14) = 21-14=7$

Step2: Evaluate $f(3)-5g(-1)$

• $f(3)$: $3\leq3<10$, so $f(3)=4(3)-9=3$.
• $g(-1)$: $-1\leq5$, so $g(-1)=0$.
• Expression: $3 - 5(0)=3$

Step3: Evaluate $0.5f(20)+g(6)$

• $f(20)$: $20\geq10$, so $f(20)=-(20)^2=-400$.
• $g(6)$: $6>5$, so $g(6)=-3(6)+16=-2$.
• Expression: $0.5(-400) + (-2) = -200-2=-202$

Answer:

a. $g(x)=

$$\begin{cases} -10, & -15 \leq x < -10 \\ -8, & -10 \leq x < -8 \\ -6, & -8 \leq x < -1 \\ 2, & -1 \leq x < 1 \\ 4, & 1 \leq x < 10 \\ 8, & 10 \leq x < 15 \end{cases}$$

$
b. $g(1)=4$, $g(-12)=-10$, $g(15)$ is undefined
c. All real numbers $x$ where $-8 \leq x < -1$

---

Problem 2