Question
If $$f\left( x \right)$$ is an invertible function and $$g\left( x \right) = 2f\left( x \right) + 5,$$ then the value of $${g^{ - 1}}\left( x \right)$$ is :
A.
$$2{f^{ - 1}}\left( x \right) - 5$$
B.
$$\frac{1}{{2{f^{ - 1}}\left( x \right) + 5}}$$
C.
$$\frac{1}{2}{f^{ - 1}}\left( x \right) + 5$$
D.
$${f^{ - 1}}\left( {\frac{{x - 5}}{2}} \right)$$
Answer :
$${f^{ - 1}}\left( {\frac{{x - 5}}{2}} \right)$$
Solution :
Replacing $$x$$ by $${g^{ - 1}}\left( x \right),$$ we get $$x = 2f\left( {{g^{ - 1}}\left( x \right)} \right) + 5$$
$$\eqalign{
& \therefore \,f\left( {{g^{ - 1}}\left( x \right)} \right) = \frac{{x - 5}}{2} \cr
& \therefore \,{g^{ - 1}}\left( x \right) = {f^{ - 1}}\left( {\frac{{x - 5}}{2}} \right) \cr} $$