Question

If $f\left(x\right)=\frac{x+2}{3x-1}$, then $f\left(f\left(x\right)\right)$ is equal to

• A. $x$
• B. $-x$
• C. $\frac{1}{x}$
• D. $-\frac{1}{x}$
• E. $0$

Answer 1

The correct option is A

$x$

Explanation for the correct option.

Find the value of $f\left(f\left(x\right)\right)$.

For the function $f\left(x\right)=\frac{x+2}{3x-1}$, the composition $f\left(f\left(x\right)\right)$ is defined as:

$\begin{array}{rcl}f\left(f\left(x\right)\right)& =& f\left(\frac{x+2}{3x-1}\right)\\ & =& \frac{{\displaystyle \frac{x+2}{3x-1}}+2}{3\times {\displaystyle \frac{x+2}{3x-1}}-1}\\ & =& \frac{x+2+2\left(3x-1\right)}{3\left(x+2\right)-\left(3x-1\right)}\\ & =& \frac{x+2+6x-2}{3x+6-3x+1}\\ & =& \frac{7x}{7}\\ & =& x\end{array}$

Hence, the correct option is A.