Question

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

$F\left(x\right)=\frac{1}{x^2+1}$

Short answer

The required functions are $f\left(x\right)=\frac{1}{x},g\left(x\right)=x+1,h\left(x\right)=x^2$.

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:

$F\left(x\right)=\frac{1}{x^2+1}$

Step 3. Express the function as composition of functions.

Since the formula for function$F\left(x\right)$ says to take square, then add 1 and then take the reciprocal.

Let,$h\left(x\right)=x^2,g\left(x\right)=x+1$ and $f\left(x\right)=\frac{1}{x}$. Then,

$$\begin{gathered} \left(f\circ g\circ h\right)\left(x\right)=f\left(g\left(h\left(x\right)\right)\right) \\ =f\left(g\left(x^2\right)\right)\left[\because h\left(x\right)=x^2\right] \\ =f\left(x^2+1\right)\left[\because g\left(x\right)=x+1\right] \\ =\frac{1}{x^2+1}\left[\because f\left(x\right)=\frac{1}{x}\right] \\ =F\left(x\right) \end{gathered}$$

Thus, the required functions are $f\left(x\right)=\frac{1}{x},g\left(x\right)=x+1,h\left(x\right)=x^2$.