Question

If $f\left(x\right)=\frac{1}{1-x},x\ne 0,1$ then the graph of the function $y=f\left[f\right\{f\left(x\right)\left\}\right]$ for $x>1$ is

• A. a straight line
• B. a circle
• C. an ellipse
• D. a pair of straight lines

Answer 1

The correct option is A a straight line

Given $f\left(x\right)=\frac{1}{1-x}$
$\Rightarrow f\left\{f\right(x\left)\right\}=\frac{1}{1-f\left(x\right)}=\frac{1}{1-\frac{1}{1-x}}=\frac{1-x}{1-x-1}=1-\frac{1}{x}$
$\therefore f\left[f\right\{f\left(x\right)\left\}\right]=\frac{1}{1-f\left\{f\right(x\left)\right\}}=\frac{1}{1-1+\frac{1}{x}}=x$, which is a straight line.