Find the Relation Between \( f(x) \) and \( f(1-x) \)
Question:
If
\[ f(x)=\frac{4^x}{4^x+2}, \qquad x\in R \]
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:
\[ f(1-x) = \frac{4^{1-x}}{4^{1-x}+2} \]
\[ = \frac{4/4^x}{4/4^x+2} \]
\[ = \frac{2}{2+4^x} \]
Now,
\[ f(x)+f(1-x) = \frac{4^x}{4^x+2} + \frac{2}{4^x+2} \]
\[ = \frac{4^x+2}{4^x+2} =1 \]
\[ \boxed{\text{Correct Answer: (c)}} \]