If f : R
Question:

If $f: \mathrm{R} \rightarrow \mathrm{R}$ be given by for all $f(x)=\frac{4^{x}}{4^{x}+2} x \in \mathrm{R}$, then

(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

 

Solution:

(c) f(x) + f(1 − x) = 1

$f(x)=\frac{4^{x}}{4^{x}+2} ; x \in \mathrm{R}$

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

$=\frac{4}{2 \times 4^{x}+4}$

$=\frac{2}{4^{x}+2}$

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

$=\frac{4^{x}+2}{4^{x}+2}=1$

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