Question

for the real-valued functions $g(x)=\frac{x+2}{x-5}$ and $h(x)=4x+1$, find the composition $g \circ h$. also, specify its domain using interval notation.
$(g \circ h)(x) = \square$
domain of $g \circ h$: $\square$

Explanation:

Step1: Define composite function

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

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

Substitute $h(x)=4x+1$ into $g(t)=\frac{t+2}{t-5}$:
$$(g\circ h)(x) = \frac{(4x+1)+2}{(4x+1)-5}$$

Step3: Simplify numerator and denominator

Simplify terms in numerator and denominator:
$$(g\circ h)(x) = \frac{4x+3}{4x-4} = \frac{4x+3}{4(x-1)}$$

Step4: Find domain restrictions

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

$4(x-1)
eq 0 \implies x
eq 1$

1. $h(x)$ must be in domain of $g$: domain of $g$ is $t

eq 5$, so $4x+1
eq 5 \implies 4x
eq 4 \implies x
eq 1$ (same restriction as above)

1. $h(x)$ is defined for all real $x$.

Combine restrictions: $x
eq 1$

Step5: Write domain in interval notation

Domain is $(-\infty,1) \cup (1,\infty)$

Answer:

$(g\circ h)(x) = \frac{4x+3}{4x-4}$
Domain of $g\circ h$: $(-\infty,1) \cup (1,\infty)$