If f(x) = x 1 + x , then f^ - 1 (x) is equal to - Vidyadip

MCQ

If $f(x) = \frac{x}{{1 + x}}$, then ${f^{ - 1}}(x)$ is equal to

A
$\frac{{(1 + x)}}{x}$
B
$\frac{1}{{(1 + x)}}$
C
$\frac{{(1 + x)}}{{(1 - x)}}$
✓
$\frac{x}{{(1 - x)}}$

✓

Answer

Correct option: D.

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

d
(d) $f(x) = \frac{x}{{1 + x}}$. Let $y = f(x) \Rightarrow x = {f^{ - 1}}(y)$

$\therefore$ $y = \frac{x}{{1 + x}} \Rightarrow y + yx = x$

==> $x = \frac{y}{{1 - y}}$

==> ${f^{ - 1}}(y) = \frac{y}{{1 - y}}$

==> ${f^{ - 1}}(x) = \frac{x}{{1 - x}}$.

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