Question

Assume that inverse of a differentiable function $f$ is denoted by $g$. Then which of the following statement hold good?

• A. If $f$ is increasing then $g$ is also increasing.
• B. If $f$ is decreasing then $g$ is increasing.
• C. The function $f$ is injective.
• D. The function $g$ is onto.

Answer 1

Answer: (A), (C), (D)

If $f$ is increasing then $g$ is also increasing.
The function $g$ is onto.
The function $f$ is injective.

Since, $g$ is an invertible function
Therefore, $f$ is injective
and since, $g$ is inverse of $f$
Therefore, $g$ is onto.
given that $f^{-1}\left(x\right)=g\left(x\right)$
$\Rightarrow x=f\left(g\left(x\right)\right)$
$\Rightarrow 1=f^{{}^{\prime }}\left(g\left(x\right)\right)g^{{}^{\prime }}\left(x\right)$
if $f$ is increasing, then $f^{\prime }\left(g\right(x\left)\right)>0$
Therefore, $g^{\prime }\left(x\right)>0$
thus $g$ is increasing.
Ans: A,C,D