Question

If $f(x+y)=f\left(x\right)+f\left(y\right)\forall x,y\in R$ and $f\left(4\right)$ is the sum to infinite terms of the series $2,1,\frac{1}{2},\frac{1}{4},\dots .$, then the image of $y=\ln (x+3)$ with respect to $y=f\left(x\right)$ is:

• A. $e^x+3$
• B. $e^{x-3}$
• C. $e^x-3$
• D. $3-e^x$

Answer 1

The correct option is C $e^x-3$

We know that,
$f(x+y)=f\left(x\right)+f\left(y\right)\Rightarrow f\left(x\right)=kx$
Also,
$f\left(4\right)=4k=\frac{2}{1-1/2}(\because \text{Sum of infinite geometric series}=\frac{a}{1-r})\Rightarrow 4k=\frac{4}{2-1}\Rightarrow k=1\therefore f\left(x\right)=x$
So, in order to find image of $y=\ln (x+3)$ with respect to $f\left(x\right)=x$
We need to find the inverse of $y=\ln (x+3)$
$\Rightarrow x+3=e^y\Rightarrow x=e^y-3$
Interchanging $x$ and $y$ we have
$y=e^x-3$
Which is the required inverse and required image.