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. $y = \frac{2x + 3}{3x + 4}$
• B. $y = \frac{2x + 3}{3x - 2}$
• C. $y = \frac{3x + 4}{4x - 5}$
• D. None of the above

Hint:

A function is self-inverse if replacing $x$ with $y$ and solving gives back the same expression.

Answer 1

The Correct Option is B

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) $y = \frac{2x + 3}{3x - 2}$}