Question

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

width="98">Fx=x−95

Short answer

The required functions are$f\left(x\right)=x^5$ and $g\left(x\right)=x-9$.

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:

$F\left(x\right)={\left(x-9\right)}^5$

Step 3. Express the function as composition of functions.

Since the formula for function$F\left(x\right)$ says to first subtract 9 and then take exponent 5.

Let,$g\left(x\right)=x-9$ and $f\left(x\right)=x^5$. Then,

$$\begin{gathered} \left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right) \\ =f\left(x-9\right)\left[\because g\left(x\right)=x-9\right] \\ ={\left(x-9\right)}^5\left[\because f\left(x\right)=x^5\right] \\ =F\left(x\right) \end{gathered}$$

Thus, the required functions are$f\left(x\right)=x^5$ and $g\left(x\right)=x-9$.