Question

A function $f: R \rightarrow R$ is such that $f(1)=2$ and $f(x+y)=f(x) \cdot f(y) \forall x, y$. The area (in square units) enclosed by the lines $2|x|+5|y| \leq 4$ expressed in terms of $f(1), f(2)$ and $f(4)$ is

• A. $\frac{f(4)}{f(1)+2 f(2)}$
• B. $\frac{f(4)}{1+f(2)}$
• C. $\frac{2 f(4)}{2 f(1)+f(2)}$
• D. $\frac{f(4)}{2 f(1)+f(2)}$

Answer 1

Answer: (B)

Given, $f(x+y)=f(x) \cdot f(y)$
$\therefore f(x)=a^{x}$
$f(1)=a^{1}=2 $
$\Rightarrow a=2$
$\therefore f(x)=2^{x}$
Area enclosed by the lines $2|x|+5|y| \leq 4$
$4\left(\frac{1}{2} \times 2 \times \frac{4}{5}\right)=\frac{16}{5}$

$=\frac{16}{1+(2)^{2}}=\frac{(2)^{4}}{1+(2)^{2}}=\frac{f(4)}{1+f(2)}$