Question

homework assignment 1.4: composition of functions
score: 7/11 answered: 7/11
question 9
the function $h(x) = (x + 7)^5$ can be expressed in the form $f(g(x))$, where $f(x) = x^5$, and $g(x)$ is defined below:
$g(x) = \square$
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Explanation:

Step1: Recall function composition

Function composition \( f(g(x)) \) means we substitute \( g(x) \) into \( f(x) \). Given \( f(x) = x^5 \), we need to find \( g(x) \) such that \( f(g(x))=(g(x))^5=(x + 7)^5 \).

Step2: Solve for \( g(x) \)

We set \( (g(x))^5=(x + 7)^5 \). By taking the fifth - root of both sides (or by inspection), we can see that \( g(x)=x + 7 \) because when we substitute \( g(x)=x + 7 \) into \( f(x)=x^5 \), we get \( f(g(x))=(x + 7)^5 \), which is equal to \( h(x) \).

Answer:

\( x + 7 \)