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
$f(x)=\frac{4^{x}}{4^{x}+2} ; x \in \mathrm{R}$
$f(1-x)=\frac{4^{1-x}}{4^{1-x}+2}$
$=\frac{4}{2 \times 4^{x}+4}$
$=\frac{2}{4^{x}+2}$
$f(x)+f(1-x)=\frac{4^{x}}{4^{x}+2}+\frac{2}{4^{x}+2}$
$=\frac{4^{x}+2}{4^{x}+2}=1$