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Question:

The set of all possible values of f(x) for which (81)^x + (81)^f(x) = 3 is

Explanation:

(81)^x + (81)^f(x) = 3 

⇒ (3)^4x + (3)^4f(x) = 3 

⇒ (3)^4x-1 + (3)^4f(x)-1 = 1   ...(1)

Now, power of a positive number is always greater than 0.

∴ (3)^4x-1 > 0 and (3)^4f(x)-1 > 0

From (1): Sum of two positive numbers is 1. This is only possible when both number are less than 1.

⇒ (3)^4f(x)-1 < 1

⇒ 4f(x) - 1 < 0

⇒ f(x) < 1/4 or 0.25

∴ f(x) ∈ (-∞, 0.25)

Hence, option (c).