If f(y) = y-1/y , find f-1(x) .

Option a: 1/1-y. Solution: To find the inverse function f-1(x) , we start by setting the function equal to an independent variable, say x : x = f(y) = y-1/y Now we solve for y in terms of x : x × y = y - 1 Rearrange the terms to group all y terms on one side: xy - y = -1 Factor out y from the left side: y(x - 1) = -1 Divide by (x-1) : y = -1/x-1 Simplify the fraction by multiplying the numerator and denominator by -1 : y = 1/1-x Therefore, the inverse function is: f-1(x) = 1/1-x If we write the inverse function using the variable y (as presented in the textbook options), we get: f-1(y) = 1/1-y Hence, Option A is the correct answer.

If $f(y) = \frac{y-1}{y}$, find $f^{-1}(x)$.

The correct answer is Option a: $\frac{1}{1-y}$. To find the inverse function $f^{-1}(x)$, we start by setting the function equal to an independent variable, say $x$: $$x = f(y) = \frac{y-1}{y}$$ Now we solve for $y$ in terms of $x$: $$x \cdot y = y - 1$$ Rearrange the terms to group all $y$ terms on one side: $$xy - y = -1$$ Factor out $y$ from the left side: $$y(x - 1) = -1$$ Divide by $(x-1)$: $$y = \frac{-1}{x-1}$$ Simplify the fraction by multiplying the numerator and denominator by $-1$: $$y = \frac{1}{1-x}$$ Therefore, the inverse function is: $$f^{-1}(x) = \frac{1}{1-x}$$ If we write the inverse function using the variable $y$ (as presented in the textbook options), we get: $$f^{-1}(y) = \frac{1}{1-y}$$ Hence, **Option A** is the correct answer.

Sets, Relations and FunctionsMCQPYQ June 23Question 1971 of 217

All Questions

If $\displaystyle f(y) = \frac{y-1}{y}$ , find $\displaystyle f^{-1}(x)$ .

Options

\displaystyle \frac{1}{1-y}

\displaystyle \frac{y}{1-y}

\displaystyle \frac{y}{y-1}

\displaystyle \frac{1}{y-1}

For any discrepancies in this question, email contact@cadada.in

Correct Answer

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

All Options:

A $\displaystyle \frac{1}{1-y}$
B $\displaystyle \frac{y}{1-y}$
C $\displaystyle \frac{y}{y-1}$
D $\displaystyle \frac{1}{y-1}$

Detailed Solution & Explanation

To find the inverse function

\displaystyle f^{-1}(x)

, we start by setting the function equal to an independent variable, say

\displaystyle x

x = f(y) = \frac{y-1}{y}

Now we solve for

\displaystyle y

in terms of

\displaystyle x

x \cdot y = y - 1

Rearrange the terms to group all

\displaystyle y

terms on one side:

xy - y = -1

Factor out

\displaystyle y

from the left side:

y(x - 1) = -1

Divide by

\displaystyle (x-1)

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

Simplify the fraction by multiplying the numerator and denominator by

\displaystyle -1

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

Therefore, the inverse function is:

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

If we write the inverse function using the variable

\displaystyle y

(as presented in the textbook options), we get:

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

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

About This Chapter: Sets, Relations and Functions

Paper

Paper 3: Quantitative Aptitude

Weightage

3-5 Marks

Key Topics

Sets, Relations, Functions

This chapter covers Sets, Relations, Functions and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 3-5 Marks weightage. Focus on understanding core concepts rather than memorizing.

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If f(y)=y−1y\displaystyle f(y) = \frac{y-1}{y}f(y)=yy−1​, find f−1(x)\displaystyle f^{-1}(x)f−1(x).