Question

If f : R → R be given by for all \[f\left( x \right) = \frac{4^x}{4^x + 2}\]  x ∈ R, then

 

Options

• (a) f(x) = f(1 − x)

• (b) f(x) + f(1 − x) = 0

• (c) f(x) + f(1 − x) = 1

• (d) f(x) + f(x − 1) = 1

   

Answer 1

(c) f(x) + f(1 − x) = 1

\[f\left( x \right) = \frac{4^x}{4^x + 2}\]

\[f (1 - x) = \frac{4^{1 - x}}{4^{1 - x} + 2}\]

\[ = \frac{4}{2 \times 4^x + 4}\]

\[ = \frac{2}{4^x + 2}\]

\[f(x) + f(1 - x) = \frac{4^x}{4^x + 2} + \frac{2}{4^x + 2} \]

\[ = \frac{4^x + 2}{4^x + 2} = 1\]

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