Answer:
The function [tex]f(x)=e^{2x}[/tex] is the inverse of the function [tex]g(x)=\frac{\ln \left(x\right)}{2}[/tex]
Step-by-step explanation:
To find the inverse function, swap x and y, and solve the resulting equation for x.
So, for the function [tex]f(x)=e^{2x}[/tex] swap the variables: [tex]y=e^{2 x}[/tex] becomes [tex]x=e^{2 y}[/tex]
Now, solve the equation [tex]x=e^{2 y}[/tex] for y.
[tex]\ln \left(x\right)=\ln \left(e^{2y}\right)\\\ln \left(x\right)=2y\ln \left(e\right)\\\ln \left(x\right)=2y\\y=\frac{\ln \left(x\right)}{2}[/tex]
For the function [tex]g(x)=\frac{\ln \left(x\right)}{2}[/tex] swap the variables: [tex]y=\frac{\ln{\left(x \right)}}{2}[/tex] becomes [tex]x=\frac{\ln{\left(y \right)}}{2}[/tex]
Now, solve the equation [tex]x=\frac{\ln{\left(y \right)}}{2}[/tex] for y.
[tex]\frac{\ln \left(y\right)}{2}=x\\\frac{2\ln \left(y\right)}{2}=2x\\\ln \left(y\right)=2x\\y=e^{2x}[/tex]
Therefore, the function [tex]f(x)=e^{2x}[/tex] is the inverse of the function [tex]g(x)=\frac{\ln \left(x\right)}{2}[/tex]
Here is the graph of the function and inverse. We can see that the graph of the inverse is a reflection of the actual function about the line y = x. This will always be the case with the graphs of a function and its inverse.
