Q 54. - Precalculus Enhanced with Graphing Utilities

Question

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

$H\left(x\right)={(1+x^2)}^3$

Step-by-Step Solution

Answer

The required functions are $f\left(x\right)=x^3; g\left(x\right)=1+x^2$

Step 1. Given information.

The given function is:

$H\left(x\right)={(1+x^2)}^3$

In the given function H takes $1+x^2$ and raises to the power 3.

Now decompose H  by raising  to the power 3.

Let's take $g\left(x\right)=1+x^2$and $f\left(x\right)=x^3$

Step 2. Find f ∘ g .

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

Substitute $g\left(x\right)=1+x^2$ in the function $f\left(g\right(x\left)\right)$,

Then the function will become $f(1+x^2)$.

Now replace x  with $1+x^2$ in $f\left(x\right)=x^3$,

$f(1+x^2)={(1+x^2)}^3{(1+x2)}^3=H\left(x\right)$

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

$f\left(x\right)=x^3; g\left(x\right)=1+x^2$