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Q.
Assume that inverse of the function f is denoted by g. Then which of the following statement
hold good?
Application of Derivatives
Solution:
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