Question

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

Answer 1

(c) f(x) + f(1 − x) = 1

$f\left(x\right)=\frac{4^x}{4^x+2}$ ; x ∈ R
$$\begin{gathered} f(1-x)=\frac{4^{1-x}}{4^{1-x}+2} \\ =\frac{4}{2\times 4^x+4} \\ =\frac{2}{4^x+2} \\ f\left(x\right)+f(1-x)=\frac{4^x}{4^x+2}+\frac{2}{4^x+2} \\ =\frac{4^x+2}{4^x+2}=1 \end{gathered}$$