Question

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

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Short answer

The value of$f\circ g\circ h$ is $\sqrt{\frac{\sqrt[3]{x}}{\sqrt[3]{x}-1}}$.

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:

$\begin{array}{c}f\left(x\right)=\sqrt{x}\\ g\left(x\right)=\frac{x}{x-1}\\ h\left(x\right)=\sqrt[3]{x}\end{array}$

Step 3. Find the required value.

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

Thus, the value of$f\circ g\circ h$ is $\sqrt{\frac{\sqrt[3]{x}}{\sqrt[3]{x}-1}}$.