Question

Composition of Three Functions Find $f\circ g\circ h$.

$\begin{array}{c}f\left(x\right)=\frac{1}{x}\\ g\left(x\right)=x^3\\ h\left(x\right)=x^2+2\end{array}$

Short answer

The value of$f\circ g\circ h$ is $\frac{1}{{\left(x^2+2\right)}^3}$.

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 functions are:

$\begin{array}{c}f\left(x\right)=\frac{1}{x}\\ g\left(x\right)=x^3\\ h\left(x\right)=x^2+2\end{array}$

Step 3. Find the required value.

$$\begin{gathered} \left(f\circ g\circ h\right)\left(x\right)=f\left(g\left(h\left(x\right)\right)\right) \\ =f\left(g\left(x^2+2\right)\right)\left[\because h\left(x\right)=x^2+2\right] \\ =f\left({\left(x^2+2\right)}^3\right)\left[\because g\left(x\right)=x^3\right] \\ =\frac{1}{{\left(x^2+2\right)}^3}\left[\because f\left(x\right)=\frac{1}{x}\right] \end{gathered}$$

Thus, the value of$f\circ g\circ h$ is $\frac{1}{{\left(x^2+2\right)}^3}$.