Question

A function $f:R\to R$ 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 $2|x|+5|y|\le 4$ expressed in terms of $f(1),f(2)$ and $f(4)$ is

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

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 $2|x|+5|y|\le 4$
$4\left(\frac{1}{2}\times 2\times \frac{4}{5}\right)=\frac{16}{5}$

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