Question

What is the inverse of the function y=3e^(4x+1)

Answer 1

to find the inverse interchange the variables and solve for y


`f^-1 (x) = - 1/4 + in(x)/4 - in(3)/4`

Answer 2

The inverse of the function
`y=3e^(4x+1)` is
`f^(-1)(x) =(\ln \left((x)/(3)\right)-1)/(4)`

Explanation:

Given the function
`y=3e^(4x+1)` we want to find the inverse function,
`f^(-1)(x)`

1. First, replace every x with a y and replace every y with an x.
2. Solve the equation from Step 1 for y.
3. Replace y with
   `f^(-1)(x)`.

Applying the above process we get:

`\mathrm{Interchange\:the\:variables}\:x\:\mathrm{and}\:y\\\\x=3e^(4y+1)\\\\\mathrm{Solve}\:x=3e^(4y+1)\:\mathrm{for}\:y\\\\3e^(4y+1)=x\\\\(3e^(4y+1))/(3)=(x)/(3)\\\\e^(4y+1)=(x)/(3)`

`\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)\\\\\ln \left(e^(4y+1)\right)=\ln \left((x)/(3)\right)\\\\\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)\\\\\left(4y+1\right)\ln \left(e\right)=\ln \left((x)/(3)\right)\\\\4y+1=\ln \left((x)/(3)\right)\\\\y=(\ln \left((x)/(3)\right)-1)/(4)\\\\f^(-1)(x) =(\ln \left((x)/(3)\right)-1)/(4)`

The inverse of the function
`y=3e^(4x+1)` is
`f^(-1)(x) =(\ln \left((x)/(3)\right)-1)/(4)`