*QUESTION:-*The function f(x) is a linear function. If f(f(x)) = f(f(f(x))) for all x, which of these is a possible form for f(x)?

**OPTIONS**

1) 2x

2) x − 8

3) ex

4) x2 – 2x + 1

5) None of these

*Solution*We assume a general expression for f(x) of the form f(x) = ax + b

We have f(f(x)) = f(ax + b) = a(ax + b) + b = a^2x + ab + b = a^2x + b(a + 1)

Similarly, f(f(f(x))) =a × f(f (x)) + b = a(a^2x + ab + b) + b = a^3x + b(a^2 + a + 1)

As f(f(x)) = f(f(f(x))), a^2x + b(a + 1) = a^3x + b(a^2 + a + 1)

Equating the coefficients of x, we get a^2 = a^3, so a = 0 or 1.

Equating the constant terms, we get b(a + 1) = b(a^2 + a + 1), so b = 0 or a = 0

The possible solutions for {a, b} are therefore {0, 0}, {0, C} and {1, 0}, where C is a real constant.

The possible solutions for f(x) are therefore f(x) = 0, f(x) = C and f(x) = x

None of the functions in the given options match with these.

Hence, option 5.

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