Question

1. given $f(x) = 3x^2 - 2x$ and $g(x) = 5x - 7$. complete the following and state the domain

a. $f^\circ g(x)$
$\

$$\begin{aligned}f^\\circ g(x)&=3(5x - 7)^2 - 2(5x - 7)\\\\&=3(25x^2 - 70x + 49)-10x + 14\\\\&=75x^2 - 210x + 147 - 10x + 14\\\\&=75x^2 - 220x + 161\\end{aligned}$$

$
domain: $\mathbb{r}$
b. $\frac{g}{f}(x)$
$\frac{g}{f}(x)=\frac{5x - 7}{3x^2 - 2x}$
domain: $x\
eq0$, $x\
eq\frac{2}{3}$

bonus +1%
given the tables of $f(x)$ and $g(x)$. find $f^\circ g(3)$

$x$	$f(x)$	$g(x)$
2	8	5
3	4	2
4	1	7
5	7	4
6	2	1

$f^\circ g(3)=$

Explanation:

Part a: $f \circ g(x)$

Step1: Substitute $g(x)$ into $f(x)$

$f(g(x)) = 3(5x-7)^2 - 2(5x-7)$

Step2: Expand $(5x-7)^2$

$(5x-7)^2 = 25x^2 - 70x + 49$

Step3: Distribute the 3

$3(25x^2 - 70x + 49) = 75x^2 - 210x + 147$

Step4: Distribute the $-2$

$-2(5x-7) = -10x + 14$

Step5: Combine like terms

$75x^2 - 210x + 147 -10x +14 = 75x^2 -220x +161$

Step6: Identify domain

The function is a polynomial, so all real numbers are allowed.

Step1: Define the rational function

$\frac{g}{f}(x) = \frac{5x-7}{3x^2 - 2x}$

Step2: Factor denominator

$3x^2 - 2x = x(3x-2)$

Step3: Find excluded values

Set denominator equal to 0: $x(3x-2)=0$ → $x=0$ or $x=\frac{2}{3}$

Step1: Find $g(3)$ from table

From the table, when $x=3$, $g(3)=2$

Step2: Find $f(g(3))=f(2)$

From the table, when $x=2$, $f(2)=8$

Answer:

$f \circ g(x) = 75x^2 - 220x + 161$
Domain: All real numbers ($\mathbb{R}$)

---

Part b: $\frac{g}{f}(x)$