Question

The area bounded by $y=f\left(x\right),x=1,x=4$ and the $x$ -axis is $12$ square units where $f\left(x\right)$ is a differentiable function such that $f\left(1\right)=3$ and $f\left(4\right)=8.$ If $f\left(x\right)$ is a bijective function which is always positive, then the area (in square units) bounded by $y=f^{-1}\left(x\right),x=3,x=8$ and the $x$ -axis is equal to

• A. $29$
• B. $32$
• C. $12$
• D. $17$

Answer 1

Answer: (D)

Since $f\left(x\right)>0\forall x\in \left[1,4\right]$
$\Rightarrow f^{-1}\left(x\right)>0\forall x\in \left[3,8\right]$

Note that the shaded area is the required area and $A_1=12.$
Let the required area be $A_2$
Also, $A_1+A_2=32-3=29$
$\Rightarrow A_2=17$ sq. units