If f≤ft( x ) = x-2/x+2, \; x -2, then what is f^{-1}(x) equal to?

Correct Answer is : 2(1+x)1−x2≤ft(1+x)/1-x1−x2(1+x)

Correct Answer is : 2(1+x)1−x2≤ft(1+x)/1-x1−x2(1+x)

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UPSC NDA I 2019 Mathematics Solved Paper

Question : 86 of 120

Marks: +1, -0

f\left( x \right) = \frac{x-2}{x+2}, \; x \ne -2

f^{-1}(x)

equal to?

Solution:

Concept:

To find the inverse function

f^{-1}(x)

, set

y = f(x)

, solve for

x

in terms of

y

, then interchange

x

and

y

Explanation:

Start with

y = \frac{x-2}{x+2}

Multiply both sides by

(x+2)

y(x+2) = x-2

Expand:

xy + 2y = x - 2

Bring

x

terms to one side:

xy - x = -2 - 2y

Factor

x

x(y-1) = -2(1+y)

Solve for

x

x = \frac{-2(1+y)}{y-1}

Simplify by multiplying numerator and denominator by

-1

x = \frac{2(1+y)}{1-y}

Now interchange

x

and

y

f^{-1}(x) = \frac{2(1+x)}{1-x}

Answer:

\displaystyle f^{-1}(x) = \frac{2(1+x)}{1-x}

, which matches option D.

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