Question

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

$H\left(x\right)=\sqrt{1+\sqrt{x}}$

Short answer

The required functions are$f\left(x\right)=\sqrt{1+x}$and $g\left(x\right)=\sqrt{x}$.

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:

$H\left(x\right)=\sqrt{1+\sqrt{x}}$

Step 3. Express the function as composition of functions.

Since the formula for function$H\left(x\right)$ says to take square root and then take square root of 1 plus of the value.

Let,$g\left(x\right)=\sqrt{x}$ and $f\left(x\right)=\sqrt{1+x}$. Then,

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

Thus, the required functions are$f\left(x\right)=\sqrt{1+x}$ and $g\left(x\right)=\sqrt{x}$.