Question

If \[f(x) = \dfrac{1}{{1 - x}},x \ne 0,1\], then find the graph of the function \[f(f(f(x))),x > 1\].
A. A circle
B. An ellipse
C. A straight line
D. A pair of straight line

Answer 1

Hint: Replace x by f(x) in the given function \[f(x) = \dfrac{1}{{1 - x}}\], then again replace x by f(x) in the function \[f(f(x)) = \dfrac{{x - 1}}{x}\]to identify the graph.

Complete step by step solution: It is given that,
\[f(x) = \dfrac{1}{{1 - x}}\]
So,
\[f(f(x)) = \dfrac{1}{{1 - f(x)}}\]
\[ = \dfrac{1}{{1 - \dfrac{1}{{1 - x}}}}\]
\[ = \dfrac{1}{{\dfrac{{1 - x - 1}}{{1 - x}}}}\]
\[ = \dfrac{{1 - x}}{{ - x}}\]
\[ = \dfrac{{x - 1}}{x}\]
And,
\[f(f(f(x))) = \dfrac{{f(x) - 1}}{{f(x)}}\]
\[ = \dfrac{{\dfrac{1}{{1 - x}} - 1}}{{\dfrac{1}{{1 - x}}}}\]
\[ = \dfrac{{\dfrac{{1 - (1 - x)}}{{1 - x}}}}{{\dfrac{1}{{1 - x}}}}\]
\[ = \dfrac{{\dfrac{x}{{1 - x}}}}{{\dfrac{1}{{1 - x}}}}\]
\[ = x\]
Therefore, the graph of \[f(f(f(x))) = x\] will be a straight line.

Option ‘C’ is correct

Additional Information: The equation of line represents a collection of sets that are in a straight line. The equation of a straight line is always a linear equation. The maximum number of variables of a straight line is 2. We take x and y as variables for an equation of a line.
 The equation of an ellipse, circle, and pair of straight lines is generally a quadratic equation.

Note: Sometime students get confused that after substituting f(x) for x how to proceed with the calculation part, so just go through the question and put the given value of f(x) and calculate to obtain the answer \[f(f(f(x))) = x\].