Question

Expressing a Function as a Composition Express the function in the form $f\circ g$.

$G\left(x\right)=\frac{x^2}{x^2+4}$

Short answer

The required functions are$f\left(x\right)=\frac{x}{x+4}$ and $g\left(x\right)=x^2$.

Step-by-step solution

Step 1. Composition of functions.

Composition of functions is defined as:

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

Step 2. Given information.

The given function is:

$G\left(x\right)=\frac{x^2}{x^2+4}$

Step 3. Express the function as composition of functions.

Since the formula for function$G\left(x\right)$ says to first square and then divide the number by the number which is 4 greater.

Let,$g\left(x\right)=x^2$ and $f\left(x\right)=\frac{x}{x+4}$. Then,

$$\begin{gathered} \left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right) \\ =f\left(x^2\right)\left[\because g\left(x\right)=x^2\right] \\ =\frac{x^2}{x^2+4}\left[\because f\left(x\right)=\frac{x}{x+4}\right] \\ =F\left(x\right) \end{gathered}$$

Thus, the required functions are$f\left(x\right)=\frac{x}{x+4}$ and $g\left(x\right)=x^2$.