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Mathematics Question on Functions

Let f:Rf : R - { 54\frac {5}{4} } →R\rightarrow R be a function defined as f(x)=5x4x+5f(x) = \frac{5x}{4x+5} . The inverse of ff is the map g:Range f →Rg : Range\,f\,\rightarrow R - { 54\frac {5}{4} } given by

A

g(y)=y5−4yg(y) = \frac{y}{5-4y}

B

g(y)=y5+4yg(y) = \frac{y}{5+4y}

C

g(y)=5y5−4yg(y) = \frac{5y}{5-4y}

D

None of theseNone\, of\, these

Answer

g(y)=5y5−4yg(y) = \frac{5y}{5-4y}

Explanation

Solution

f : R -\left\\{\frac{5}{4}\right\\}\rightarrow R and f(x)=5x4x+5f \left(x\right)=\frac{5x}{4x+5}
Let y=5x4x+5y=\frac{5x}{4x+5}
⇒4xy+5y=5x\Rightarrow 4xy+5y=5x
⇒x(5−4y)=5y\Rightarrow x\left(5-4y\right)=5y
⇒x=5y(5−4y)=f−1(y)\Rightarrow x=\frac{5y}{\left(5-4y\right)}=f^{-1}\left(y\right)
⇒f−1(x)=5x5−4x\Rightarrow f^{-1}\left(x\right)=\frac{5x}{5-4x}
or g(y)=5y5−4yg\left(y\right)=\frac{5y}{5-4y}
which is the inverse of ff is the map g.
Range f \rightarrow R-\left\\{\frac{5}{4}\right\\}