Q 56. - Precalculus Enhanced with Graphing Utilities

Question

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

$H\left(x\right)=\sqrt{1-x^2}$

Step-by-Step Solution

Answer

The required functions are $f\left(x\right)=\sqrt{x}; g\left(x\right)=1-x^2$.

Step 1. Given information.

The given function is:

$H\left(x\right)=\sqrt{1-x^2}$

In the given function H  takes $1-x^2$ and raises to the power $\frac{1}{2}$.

Now decompose H  by raising $g\left(x\right)=1-x^2$ to the power $\frac{1}{2}$.

Let's take $g\left(x\right)=1-x^2$and $f\left(x\right)=\sqrt{x}$.

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 $f\left(x\right)=\sqrt{x}$,

$$\begin{gathered} f(1-x^2)=\sqrt{1-x^2} \\ \sqrt{1-x^2}=H\left(x\right) \end{gathered}$$

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

$f\left(x\right)=\sqrt{x}; g\left(x\right)=1-x^2$