Question

Assume that inverse of the 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 $f$ and $g$ are inverse then $(f\circ g)(x)=x$
$f^{\prime }[g(x)]g^{\prime }(x)=1$
if $f$ is increasing $\Rightarrow f^{\prime }>0\Rightarrow$ sign of $g^{\prime }$ is also $+ve\Rightarrow$(A) is correct
If $f$ is decreasing $\Rightarrow f^{\prime }<0\Rightarrow$ sign of $g^{\prime }$ is $-ve\Rightarrow$(B) is false
since $f$ has an inverse $\Rightarrow f$ is bijective $\Rightarrow$ fis injective$\Rightarrow$(C) is correct
inverse of a bijective mapping is bijective
$\Rightarrow$ gis also bijective $\Rightarrow$ g is onto $\Rightarrow$ (D) is correct