Question

Inverse Function In Exercise,analytically show that the functions are inverse functions.Then use the graphing utility to show this graphically.

f(x) = e^2x
g(x) = In(x)1/2

Answer 1

The function
`f(x)=e^(2x)` is the inverse of the function
`g(x)=(\ln \left(x\right))/(2)`

Explanation:

To find the inverse function, swap x and y, and solve the resulting equation for x.

So, for the function
`f(x)=e^(2x)` swap the variables:
`y=e^(2 x)` becomes
`x=e^(2 y)`

Now, solve the equation
`x=e^(2 y)` for y.

`\ln \left(x\right)=\ln \left(e^(2y)\right)\\\ln \left(x\right)=2y\ln \left(e\right)\\\ln \left(x\right)=2y\\y=(\ln \left(x\right))/(2)`

For the function
`g(x)=(\ln \left(x\right))/(2)` swap the variables:
`y=(ln(\left(x \right)))/(2)` becomes
`x=(ln(\left(y \right)))/(2)`

Now, solve the equation
`x=(ln(\left(y \right)))/(2)` for y.

`(\ln \left(y\right))/(2)=x\\(2\ln \left(y\right))/(2)=2x\\\ln \left(y\right)=2x\\y=e^(2x)`

Therefore, the function
`f(x)=e^(2x)` is the inverse of the function
`g(x)=(\ln \left(x\right))/(2)`

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.