Q 53. - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

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

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

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

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

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