MCQ
If $f : R \rightarrow R$ be given by for all $\text{f(x)}=\frac{4^{\text{x}}}{4^{\text{x}}+2}\text{ x }\in\text{R},$ then:
- A$f(x) = f(1 - x)$
- B$f(x) + f(1 - x) = 0$
- ✓$f(x) + f(1 - x) = 1$
- D$f(x) + f(x - 1) = 1$
✓
Answer
Correct option: C.
$f(x) + f(1 - x) = 1$
$\text{f(x)}=\frac{4^{\text{x}}}{4^{\text{x}}+2}\text{ x }\in\text{R},$
$\text{f}\big(\text{1}-\text{x}\big)=\frac{4^{1-\text{x}}}{4^{1-\text{x}}+2}$
$=\frac{4}{2\times4^{\text{x}}+4}$
$=\frac{2}{4^{\text{x}}+2}$
$\text{f(x)}+\text{f}(1-\text{x})=\frac{4^{\text{x}}}{4^{\text{x}}+2}+\frac{2}{4^{\text{x}}+2}$
$=\frac{4^{\text{x}}+2}{4^{\text{x}}+2}=1$
$\text{f}\big(\text{1}-\text{x}\big)=\frac{4^{1-\text{x}}}{4^{1-\text{x}}+2}$
$=\frac{4}{2\times4^{\text{x}}+4}$
$=\frac{2}{4^{\text{x}}+2}$
$\text{f(x)}+\text{f}(1-\text{x})=\frac{4^{\text{x}}}{4^{\text{x}}+2}+\frac{2}{4^{\text{x}}+2}$
$=\frac{4^{\text{x}}+2}{4^{\text{x}}+2}=1$
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