9 multiple answer 10 points\nif $f(x) = x - 6$ and $g(x) = \\frac{1}{x+2}$, find $(g \\circ f)(x)$ and its…

9 multiple answer 10 points\nif $f(x) = x - 6$ and $g(x) = \\frac{1}{x+2}$, find $(g \\circ f)(x)$ and its domain. select all that apply. please do not guess, as points are deducted for incorrect answers on this type of problem.\ndomain: $(-\\infty, 4) \\cup (4, \\infty)$\n$(g \\circ f)(x) = \\frac{1}{x-6} + 2$\ndomain: $(-\\infty, 6) \\cup (6, \\infty)$\n$(g \\circ f)(x) = \\frac{1}{x-4}$

9 multiple answer 10 points\nif $f(x) = x - 6$ and $g(x) = \\frac{1}{x+2}$, find $(g \\circ f)(x)$ and its domain. select all that apply. please do not guess, as points are deducted for incorrect answers on this type of problem.\ndomain: $(-\\infty, 4) \\cup (4, \\infty)$\n$(g \\circ f)(x) = \\frac{1}{x-6} + 2$\ndomain: $(-\\infty, 6) \\cup (6, \\infty)$\n$(g \\circ f)(x) = \\frac{1}{x-4}$

Answer

Explanation:

Step1: Compute composite function $(g\circ f)(x)$

Substitute $f(x)$ into $g(x)$: $$(g\circ f)(x) = g(f(x)) = g(x-6) = \frac{1}{(x-6)+2}$$ Simplify the denominator: $$(g\circ f)(x) = \frac{1}{x-4}$$

Step2: Find domain of $(g\circ f)(x)$

Denominator cannot be zero: $$x-4 \neq 0 \implies x \neq 4$$ Also, ensure $f(x)$ is defined (it is defined for all real $x$), so the domain is all real numbers except $4$, written as $(-\infty, 4) \cup (4, \infty)$.

Answer:

$(g\circ f)(x) = \frac{1}{x-4}$ Domain: $(-\infty, 4) \cup (4, \infty)$