Question

homework assignment 1.4: composition of functions
score: 8/11 answered: 8/11
question 10
the function $h(x) = \frac{1}{x - 5}$ can be expressed in the form $f(g(x))$, where $g(x) = (x - 5)$, and $f(x)$ is defined as:
$f(x) = \square$
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Explanation:

Step1: Recall function composition

Function composition \( f(g(x)) \) means we substitute \( g(x) \) into \( f(x) \). We know \( h(x) = f(g(x))=\frac{1}{x - 5} \) and \( g(x)=x - 5 \). So we need to find \( f(x) \) such that when we replace \( x \) in \( f(x) \) with \( g(x)=x - 5 \), we get \( \frac{1}{x - 5} \).

Step2: Determine \( f(x) \)

Let \( u = g(x)=x - 5 \). Then \( h(x)=\frac{1}{u} \). So if we let \( f(u)=\frac{1}{u} \), then replacing \( u \) with \( x \) (since the variable name doesn't matter), we get \( f(x)=\frac{1}{x} \).

Answer:

\( \frac{1}{x} \)