Q 45. - Precalculus Enhanced with Graphing Utilities

Question

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

$f\left(x\right)=2x; g\left(x\right)=\frac{1}{2}x$

Step-by-Step Solution

Answer

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

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

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

Step 1. Given information

The given composite function is:

$$\begin{gathered} f\left(x\right)=2x \\ g\left(x\right)=\frac{1}{2}x \end{gathered}$$

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$g\left(x\right)=\frac{1}{2}x$ in the function $f\left(g\right(x\left)\right)$,

Then the function will become $f\left(\frac{1}{2}x\right)$.

Now replace x with $\frac{1}{2}x$ in $f\left(x\right)=2x$,

$$\begin{gathered} f\left(\frac{1}{2}x\right)=2\left(\frac{1}{2}x\right) \\ =x \end{gathered}$$

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 2x in the function $g\left(f\right(x\left)\right)$.

Then the function will become $g\left(2x\right)$,

$$\begin{gathered} g\left(2x\right)=\frac{1}{2}\left(2x\right) \\ =x \end{gathered}$$

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

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