Set, Relations and FunctionsPYQ June 23Question 1971 of 217
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If f(y)=y−1y\displaystyle f(y) = \frac{y-1}{y}, find f−1(x)\displaystyle f^{-1}(x).

Options

A11−y\displaystyle \frac{1}{1-y}
By1−y\displaystyle \frac{y}{1-y}
Cyy−1\displaystyle \frac{y}{y-1}
D1y−1\displaystyle \frac{1}{y-1}
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Correct Answer

✅ Option a — 11−y\displaystyle \frac{1}{1-y}

All Options:

  • A11−y\displaystyle \frac{1}{1-y}
  • By1−y\displaystyle \frac{y}{1-y}
  • Cyy−1\displaystyle \frac{y}{y-1}
  • D1y−1\displaystyle \frac{1}{y-1}

Detailed Solution & Explanation

To find the inverse function f−1(x)\displaystyle f^{-1}(x), we start by setting the function equal to an independent variable, say x\displaystyle x:
x=f(y)=y−1yx = f(y) = \frac{y-1}{y}
Now we solve for y\displaystyle y in terms of x\displaystyle x:
x⋅y=y−1x \cdot y = y - 1
Rearrange the terms to group all y\displaystyle y terms on one side:
xy−y=−1xy - y = -1
Factor out y\displaystyle y from the left side:
y(x−1)=−1y(x - 1) = -1
Divide by (x−1)\displaystyle (x-1):
y=−1x−1y = \frac{-1}{x-1}
Simplify the fraction by multiplying the numerator and denominator by −1\displaystyle -1:
y=11−xy = \frac{1}{1-x}
Therefore, the inverse function is:
f−1(x)=11−xf^{-1}(x) = \frac{1}{1-x}
If we write the inverse function using the variable y\displaystyle y (as presented in the textbook options), we get:
f−1(y)=11−yf^{-1}(y) = \frac{1}{1-y}

Hence, **Option A** is the correct answer.

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