Q 60. - Precalculus Enhanced with Graphing Utilities

Question

If $f\left(x\right)=\frac{x+1}{x-1}$find $(f o f)\left(x\right)$.

Step-by-Step Solution

Answer

The value of$(f o f)\left(x\right)=x$.

Step 1. Given information

$f\left(x\right)=\frac{x+1}{x-1}$and the function$(f o f)\left(x\right)=f\left(f\right(x\left)\right)$

Step 2. The given function written as.

$f\left(f\right(x\left)\right)=\frac{{\displaystyle \frac{x+1}{x-1}}+1}{{\displaystyle \frac{x+1}{x-1}}-1}$

Step 3. Simplify the given function.

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