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Mathematics Question on Differentiability

If f:R→R,g:R→Rf: R \rightarrow R, g: R \rightarrow R are defined by f(x)=5 x−3,g(x)=x2+3f(x)=5\, x-3, g(x)=x^{2}+3, then gof−1(3)g o f^{-1}(3) is equal to

A

253\frac{25}{3}

B

11125\frac{111}{25}

C

925\frac{9}{25}

D

25111\frac{25}{111}

Answer

11125\frac{111}{25}

Explanation

Solution

Given, f(x)=5x−3f(x)=5 x-3
Let y=5x−3y=5 x-3
⇒y+3=5x\Rightarrow y+3=5 x
⇒x=y+35\Rightarrow x=\frac{y+3}{5}
∴f−1(y)=y+35\therefore f^{-1}(y)=\frac{y+3}{5}
⇒f−1(x)=x+35\Rightarrow f^{-1}(x)=\frac{x+3}{5}
and g(x)=x2+3g(x)=x^{2}+3
Now, gof−1(3)=g[f−1(3)]g o f^{-1}(3)=g\left[f^{-1}(3)\right]
=g(3+35)=g(65)=g\left(\frac{3+3}{5}\right)=g\left(\frac{6}{5}\right)
⇒g(65)=(6)2(5)2+3=3625+3=11125\Rightarrow g\left(\frac{6}{5}\right)=\frac{(6)^{2}}{(5)^{2}}+3 =\frac{36}{25}+3=\frac{111}{25}