Question

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

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

Answer 1

Answer: (C)

a straight line

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

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

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

$=\frac{1}{1-\frac{1}{\frac{1-x-1}{1-x}}}=\frac{1}{1-\frac{(1-x)}{-x}}=\frac{1}{1+\frac{1-x}{x}}=\frac{1}{\frac{x+1-x}{x}}=\frac{x}{1}$

$\Rightarrow y=x$

Which represents a straight line. [C].