Question

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

$f\left(x\right)=4-3x;g\left(x\right)=\frac{1}{3}(4-x)$

Short 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-by-step solution

Step 1. Given information.

The given composite function is:

$$\begin{gathered} f\left(x\right)=4-3x \\ g\left(x\right)=\frac{1}{3}(4-x) \end{gathered}$$

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 $g\left(x\right)=\frac{1}{3}(4-x)$in the function $f\left(g\right(x\left)\right)$,

Then the function will become $f\left(\frac{1}{3}\right(4-x\left)\right)$.

$$\begin{gathered} f\left(\frac{1}{3}\right(4-x\left)\right)=4-3\left(\frac{1}{3}\right(4-x\left)\right) \\ =4-3\left(\frac{4-x}{3}\right) \\ =4-4+x \\ =x \end{gathered}$$

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$f\left(x\right)=4-3x$in the function$g\left(f\right(x\left)\right)$,

localid="1646305216024" $$\begin{gathered} (g\circ f)\left(x\right)=g\left(f\right(x\left)\right) \\ =g(4-3x) \\ =\frac{1}{3}(4-(4-3x\left)\right) \\ =\frac{1}{3}(4-4+3x) \\ =x \end{gathered}$$

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

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