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

Answer:

The inverse of the function $y=3e^{4x+1}$  is $f^{-1}(x) =\frac{\ln \left(\frac{x}{3}\right)-1}{4}$

Step-by-step 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:

$\mathrm{Interchange\:the\:variables}\:x\:\mathrm{and}\:y\\\\x=3e^{4y+1}\\\\\mathrm{Solve}\:x=3e^{4y+1}\:\mathrm{for}\:y\\\\3e^{4y+1}=x\\\\\frac{3e^{4y+1}}{3}=\frac{x}{3}\\\\e^{4y+1}=\frac{x}{3}$

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

The inverse of the function $y=3e^{4x+1}$  is $f^{-1}(x) =\frac{\ln \left(\frac{x}{3}\right)-1}{4}$