Question:
A function can sometimes reflect on itself, i.e. if $y = f(x)$, then $x = f(y)$. Both of them retain the same structure and form. Which of the following functions has this property?
A function can sometimes reflect on itself, i.e. if $y = f(x)$, then $x = f(y)$. Both of them retain the same structure and form. Which of the following functions has this property?
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A function is self-inverse if replacing $x$ with $y$ and solving gives back the same expression.
Updated On: Jul 20, 2026
- $y = \frac{2x + 3}{3x + 4}$
- $y = \frac{2x + 3}{3x - 2}$
- $y = \frac{3x + 4}{4x - 5}$
- None of the above
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The Correct Option is B
Solution and Explanation
We are told that if $y = f(x)$, then $x = f(y)$. This means the function is its own inverse:
\[
f^{-1}(x) = f(x)
\]
Step 1: Test option (a)
Given: $y = \frac{2x + 3}{3x + 4}$
Interchange $x$ and $y$:
\[
x = \frac{2y + 3}{3y + 4} \quad \Rightarrow \quad x(3y + 4) = 2y + 3
\]
\[
3xy + 4x = 2y + 3 \quad \Rightarrow \quad 3xy - 2y = 3 - 4x
\]
\[
y(3x - 2) = 3 - 4x \quad \Rightarrow \quad y = \frac{3 - 4x}{3x - 2}
\]
This is not the same as the original function $\frac{2x + 3}{3x + 4}$, so (a) is rejected.
Step 2: Test option (b)
Given: $y = \frac{2x + 3}{3x - 2}$
Swap $x$ and $y$:
\[
x = \frac{2y + 3}{3y - 2} \quad \Rightarrow \quad x(3y - 2) = 2y + 3
\]
\[
3xy - 2x = 2y + 3 \quad \Rightarrow \quad 3xy - 2y = 2x + 3
\]
\[
y(3x - 2) = 2x + 3 \quad \Rightarrow \quad y = \frac{2x + 3}{3x - 2}
\]
This matches the original function exactly — so (b) satisfies the property.
Step 3: Conclusion
The function in (b) is self-inverse.
\fbox{Final Answer: (b) $y = \frac{2x + 3}{3x - 2}$}
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