Q 47. - Precalculus Enhanced with Graphing Utilities

Question

In Problems 45–52, show that $(f\circ g)\left(x\right)=(g\circ f)\left(x\right)=x$

$f\left(x\right)=x^3; g\left(x\right)=\sqrt[3]{x}$

Step-by-Step Solution

Answer

$$\begin{gathered} (f\circ g)\left(x\right)=x \\ (g\circ f)\left(x\right)=x \end{gathered}$$

Therefore, $(f\circ g)\left(x\right)=(g\circ f)\left(x\right)=x$.

Step 1. Given information.

The given composite function is:

$$\begin{gathered} f\left(x\right)=x^3 \\ g\left(x\right)=\sqrt[3]{x} \end{gathered}$$

When we are given two functions f and g, the composite function which is denoted by $f\circ g$ is defined by $(f\circ g)\left(x\right)=f\left(g\right(x\left)\right)$.

Step 2. Find ( f ∘ g ) ( x ) .

$(f\circ g)\left(x\right)=f\left(g\right(x\left)\right)$

Now substitute $g\left(x\right)=\sqrt[3]{x}$ in the function $f\left(g\right(x\left)\right)$,

Then the function will become $f\left(\sqrt[3]{x}\right)$,

$$\begin{gathered} f\left(\sqrt[3]{x}\right)=({\left(\sqrt[3]{x}\right))}^3 \\ =x \end{gathered}$$

Therefore, $(f\circ g)\left(x\right)=x$.

Step 3. Find ( g ∘ f ) ( x ) .

$(g\circ f)\left(x\right)=g\left(f\right(x\left)\right)$

Substitute $f\left(x\right)=x^3$ in $g\left(f\right(x\left)\right)$,

$$\begin{gathered} (g\circ f)\left(x\right)=g\left(f\right(x\left)\right) \\ =g\left(x^3\right) \\ =\sqrt[3]{x^3} \\ =x \end{gathered}$$

$(g\circ f)\left(x\right)=x$

It is shown that $(f\circ g)\left(x\right)=(g\circ f)\left(x\right)=x$.