Question

Assume that inverse of the function $f$ is denoted by $g$, then which of the following statement hold(s) 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)

The function $g$ is onto

If $f$ and $g$ are inverse, then $\left(fog\right)\left(x\right)=x$
$f^{\prime }\left[g\right(x\left)\right]g^{\prime }\left(x\right)=1$
If $f$ is increasing
$\Rightarrow f^{\prime }>0$
$\Rightarrow$ Sign of $g^{\prime }$ is also positive.
Therefore, option $\left(a\right)$ is correct.
If $f$ is decreasing
$\Rightarrow f^{\prime }<0$
$\Rightarrow$ Sign of $g^{\prime }$ is negative.
Therefore, option $\left(b\right)$ is false.
Since, $f$ has an inverse.
$f$ is bijective
$\Rightarrow f$ is injective Therefore, option $\left(c\right)$ is correct. Inverse of a bijective mapping is bijective.
$\Rightarrow g$ is also bijective $g$ is onto
Therefore, option $\left(d\right)$ is correct.
Hence, $\left(a\right),\left(c\right)$ and $\left(d\right)$ are the correct.