Question

find the inverse of g:R- R,g(x)=4e^2x+3 and the domain of the g^-1 (X)

Answer 1

Answer: log( square root ((x-3)/4))

Answer 2

The inverse of g(x) is g^-1(x) = $ln\sqrt{\frac{x-3}{4} }$ and domain is (3,∞).

• let g(x)=y. hence y = $4e^{2x}+3$

• Hence, y-3 = $4e^{x}$

• or, $e^{2x}$= $\frac{y-3}{4}$

• or, $e^{x}$=$\sqrt[]{\frac{y-3}{4} }$

• or, x =$ln\sqrt{\frac{y-3}{4} }$

• now as g(x)= y,then x = g^-1(y) = $ln\sqrt{\frac{y-3}{4} }$

• hence g^-1(x) = $ln\sqrt{\frac{x-3}{4} }$

• As we know the domain of logarithm is positive real number then so the domain of g^-1(x) is (3,∞)