If $$f\left( x \right) = x - {x^2} + {x^3} - {x^4} + .....\,{\text{to }}\infty {\text{ for }}\left| x \right| < 1,$$ then $${f^{ - 1}}\left( x \right) = ?$$
A.
$$\frac{x}{{1 + x}}$$
B.
$$\frac{x}{{1 - x}}$$
C.
$$\frac{{1 - x}}{x}$$
D.
$$\frac{1}{x}$$
Answer :
$$\frac{x}{{1 - x}}$$
Solution :
$$\eqalign{
& {\text{Given }}f\left( x \right) = x - {x^2} + {x^3} - {x^4} + .....\,{\text{to }}\infty \cr
& \Rightarrow y = \frac{x}{{1 + x}}\,\,\,\,\,\left( {{\text{Infinite}}\,{\text{G}}{\text{.P}}{\text{.}}} \right) \cr
& \Rightarrow y + xy = x \cr
& \Rightarrow y = x\left( {1 - y} \right) \cr
& \Rightarrow x = \frac{y}{{1 - y}} \cr
& \Rightarrow {f^{ - 1}}\left( y \right) = \frac{y}{{1 - y}} \cr
& \Rightarrow {f^{ - 1}}\left( x \right) = \frac{x}{{1 - x}} \cr} $$
Releted MCQ Question on Calculus >> Sets and Relations
Releted Question 1
If $$X$$ and $$Y$$ are two sets, then $$X \cap {\left( {X \cup Y} \right)^c}$$ equals.