Question

If $f$ is an invertible function given by $f\left(x\right)=\ln \left(x\right)+4x-8,$ then the value of $f^{-1}(-4)$ is

• A. $1$
• B. $0$
• C. $-1$
• D. $2$

Answer 1

The correct option is A $1$

Let $a=f^{-1}(-4)$. Then $f\left(a\right)=f\left(f^{-1}\right(-4\left)\right)=-4$ $[\because f\left(f^{-1}\right(x\left)\right)=x]$ .
We now need to find $a$ such that $f\left(a\right)=-4$
$\Rightarrow \ln \left(a\right)+4a-8=-4$
$\Rightarrow \ln \left(a\right)=4-4a$
The above equation can be solved graphically as the $x$ coordinate of the point of intersection of graphs for $y=\ln \left(x\right)$ and $y=4-4x$ as shown below.


The intersection of two graphs is at $x=1.$
Hence, $f^{-1}(-4)=1$