Question

Hello, May I please get some assistance with this homework question? I posted an image below. Parts A and B are already completed. Q2 (Part C.)

Answer 1

SOLUTION

The given functions are

$f(x)=4x+3,g(x)=x^2$

Recall the rule of composite function

$f\circ g=f(g(x))$

a. f o g

This can be rewritten as

$f\circ g=f(g(x))$

Substitute the value of g(x) into the expression

This gievs

$f(g(x))=f(x^2)$

Substitute x² for x into the expression

$f(x^2)=4x^2+3$

Therefore fog is

$4x^2+3$

The domain of fog is all real numbers

Following the same procedure it follows

b.

$\begin{gathered} \text{gof}=g(f(x)) \\ =g(4x+3) \\ =(4x+3)^2 \end{gathered}$

The domain of gof is all real numbers

c. fof

$\begin{gathered} f\circ f=f(f(x)) \\ =f(4x+3) \\ =4(4x+3)+3 \\ =16x+12+3 \\ =16x+15 \end{gathered}$

The domain of fof is all real numbers

d. gog

$\begin{gathered} g\circ g=g(g(x)) \\ =g(x^2) \\ =(x^2)^2 \\ =x^4 \end{gathered}$

The domain of gog is all real numbers