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

Given,

$$81^x+81^{f\left(x\right)}=3$$

Now, $$81^{f\left(x\right)}=3-81^x$$

or, $$81^{f\left(x\right)}=3-3^{4x}$$

Now, $$81^{f\left(x\right)}$$ is an exponential term which is always greater than zero

So, $$3-3^{4x}>0$$

or, $$3>3^{4x}$$

or, $$4x<1$$

or, $$x<0.25$$

So, x will lie in the interval $$(-\infty, 0.25)$$

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