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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?

Explanation:

For each of the given expressions, you may have to simplify and express x in terms of y and hence verify for which one does the form & structure remain the same.

In general, any function of the form y = $\frac{a x + b}{b x - a}$ reflects on to itself as we can rearrange it as

x = $\frac{a y + b}{b y - a}$

Let us check option (b).

y = f(x)

⇒ y = $\frac{2 x + 3}{3 x - 2}$

⇒ 3xy - 2y = 2x + 3

⇒ 3xy - 2x = 2y + 3

⇒ x(3y - 2) = (2y + 3)

⇒ x = $\frac{2 y + 3}{3 y - 2}$

∴ x = f(y)

Hence, the answer is (b).

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