Question

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

$G\left(x\right)=\frac{2}{{\left(3+\sqrt{x}\right)}^2}$

Short answer

The required functions are $f\left(x\right)=\frac{2}{x},g\left(x\right)=x^2,h\left(x\right)=3+\sqrt{x}$.

Step-by-step solution

Step 1. Composition of functions.

Composition of functions is defined as:

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

Step 2. Given information.

The given function is:

$G\left(x\right)=\frac{2}{{\left(3+\sqrt{x}\right)}^2}$

Step 3. Express the function as composition of functions.

Since the formula for function$G\left(x\right)$ says to take sum of 3 and square root, then take square and then divide 2 by the number.

Let,$h\left(x\right)=3+\sqrt{x},g\left(x\right)=x^2$ and $f\left(x\right)=\frac{2}{x}$. Then,

$$\begin{gathered} \left(f\circ g\circ h\right)\left(x\right)=f\left(g\left(h\left(x\right)\right)\right) \\ =f\left(g\left(3+\sqrt{x}\right)\right)\left[\because h\left(x\right)=3+\sqrt{x}\right] \\ =f\left({\left(3+\sqrt{x}\right)}^2\right)\left[\because g\left(x\right)=x^2\right] \\ =\frac{2}{{\left(3+\sqrt{x}\right)}^2}\left[\because f\left(x\right)=\frac{2}{x}\right] \\ =G\left(x\right) \end{gathered}$$

Thus, the required functions are $f\left(x\right)=\frac{2}{x},g\left(x\right)=x^2,h\left(x\right)=3+\sqrt{x}$.