To express the function [tex]\( h(x) = \frac{1}{x-4} \)[/tex] in the form [tex]\( f(g(x)) \)[/tex], where [tex]\( g(x) = x-4 \)[/tex], we need to identify what [tex]\( f(x) \)[/tex] should be.
Let's proceed step-by-step to find [tex]\( f(x) \)[/tex]:
1. Identify [tex]\( g(x) \)[/tex]:
We are given that [tex]\( g(x) = x - 4 \)[/tex].
2. Express [tex]\( h(x) \)[/tex] using [tex]\( g(x) \)[/tex]:
We know [tex]\( h(x) = \frac{1}{x-4} \)[/tex].
Notice that [tex]\( x-4 \)[/tex] is actually [tex]\( g(x) \)[/tex]. So we can rewrite [tex]\( h(x) \)[/tex] as:
[tex]\[
h(x) = \frac{1}{g(x)}
\][/tex]
3. Determine [tex]\( f(y) \)[/tex]:
We need to find a function [tex]\( f(y) \)[/tex] such that [tex]\( f(g(x)) = h(x) \)[/tex].
From the equation above, it follows that:
[tex]\[
f(g(x)) = \frac{1}{g(x)}
\][/tex]
Since [tex]\( g(x) \)[/tex] is just a placeholder representing [tex]\( x-4 \)[/tex], we can say:
[tex]\[
f(y) = \frac{1}{y}
\][/tex]
where [tex]\( y = g(x) \)[/tex].
Therefore, the function [tex]\( h(x) = \frac{1}{x-4} \)[/tex] can be expressed as [tex]\( f(g(x)) \)[/tex] where [tex]\( g(x) = x-4 \)[/tex] and:
[tex]\[
f(x) = \frac{1}{x}
\][/tex]
