Question

In Problems 53–58, find functionsf and g so that$f\circ g=H$.

$H\left(x\right)={(2x+3)}^4$

Short answer

The required functions are$f\left(x\right)=x^4;g\left(x\right)=2x+3$.

Step-by-step solution

Step 1. Given information. 

The given function is:

$H\left(x\right)={(2x+3)}^4$

In the given function H takes $2x+3$and raises to the power 4.

Now decompose H by raising $g\left(x\right)=2x+3$to the power 4.

Let's take role="math" localid="1646318436355" $g\left(x\right)=2x+3$and$f\left(x\right)=x^4$.

Step 2. Find f∘g.

$(f\circ g)\left(x\right)=f\left(g\right(x\left)\right)$

Substitute $g\left(x\right)=2x+3$in the function $f\left(g\right(x\left)\right)$,

Then the function will become $f(2x+3)$.

Now replace x with$2x+3$in$f\left(x\right)=x^4$,

localid="1646319035134" $$\begin{gathered} f(2x+3)={(2x+3)}^4 \\ {(2x+3)}^4=H\left(x\right) \end{gathered}$$

As we can see that $f\circ g=H$, therefore the values of the function that we assumed are correct.

$f\left(x\right)=x^4;g\left(x\right)=2x+3$