Question

for the real - valued functions ( g(x)=\frac{x - 6}{x + 3} ) and ( h(x)=4x + 5 ), find the composition ( gcirc h ). also, specify its domain using interval notation.

Explanation:

Step1: Define composition $g\circ h$

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

Step2: Substitute $h(x)$ into $g$

$g(h(x)) = \frac{h(x)-6}{h(x)+3} = \frac{(4x+5)-6}{(4x+5)+3}$

Step3: Simplify numerator and denominator

Numerator: $4x+5-6 = 4x-1$
Denominator: $4x+5+3 = 4x+8$
So $g\circ h(x) = \frac{4x-1}{4x+8}$

Step4: Simplify denominator factor

$\frac{4x-1}{4(x+2)}$

Step5: Find domain restrictions

1. Denominator of $g\circ h$ cannot be 0:

$4x+8
eq 0 \implies x
eq -2$

1. Input to $g$ must be in domain of $g$: domain of $g$ is $x

eq -3$, so $h(x)
eq -3$:
$4x+5
eq -3 \implies 4x
eq -8 \implies x
eq -2$
Combined restriction: $x
eq -2$

Step6: Write domain in interval notation

$(-\infty, -2) \cup (-2, \infty)$

Answer:

The composition is $\boldsymbol{\frac{4x-1}{4x+8}}$ (or simplified as $\boldsymbol{\frac{4x-1}{4(x+2)}}$), and its domain is $\boldsymbol{(-\infty, -2) \cup (-2, \infty)}$