Question

Find f(x) and g(x) such that h(x) = (f ∘ g)(x).

Answer 1

We have here a case of composition between two functions, and we have the resulting function for the composition of two functions, namely, f(x) and g(x).

We need to find them such the result of that composition is:

`h(x)=\lvert{3x+7}\rvert`

Now, we can see that we have two functions: one is the function absolute value:

`f(x)=\lvert{x}\rvert`

And we have another linear function, represented by:

`g(x)=3x+7`

We also have to remember that the composition between two functions is an operation that is equal to:

`(f\circ g)(x)=f(g(x))`

That is, instead of writing the value for x in the first function, we have to substitute, x, with the expression of the second function. Then, if we have:

`\begin{gathered} f(x)=\lvert{x}\rvert \\ \\ g(x)=3x+7 \\ \\ (f\circ g)(x)=f(g(x))=\lvert{3x+7}\rvert=h(x) \\ \\ f(g(x))=h(x)=\lvert{3x+7}\rvert \end{gathered}`

Therefore, in summary, we have that both functions are:

`\begin{gathered} f(x)=\lvert{x}\rvert,g(x)=3x+7 \\ \\ h(x)=|3x+7| \end{gathered}`

[Last option.]