Math Problem Statement

if f(x) = e^(4x-3), find f^-1(x)

Solution

To find the inverse function f1(x)f^{-1}(x) for the given function f(x)=e4x3f(x) = e^{4x-3}, follow these steps:

  1. Start with the function:
    y=f(x)=e4x3y = f(x) = e^{4x-3}

  2. Swap xx and yy to solve for the inverse function:
    x=e4y3x = e^{4y-3}

  3. Take the natural logarithm (ln) of both sides to solve for yy:
    ln(x)=4y3\ln(x) = 4y - 3

  4. Isolate yy:
    4y=ln(x)+34y = \ln(x) + 3 y=ln(x)+34y = \frac{\ln(x) + 3}{4}

So, the inverse function is: f1(x)=ln(x)+34f^{-1}(x) = \frac{\ln(x) + 3}{4}

Would you like more details or have any questions?


Related Questions:

  1. What is the domain of f1(x)f^{-1}(x)?
  2. How do you determine if f(x)f(x) is one-to-one?
  3. What is the range of the original function f(x)f(x)?
  4. How do you differentiate f1(x)f^{-1}(x)?
  5. Can this method be used for functions with different exponents, such as f(x)=e2x5f(x) = e^{2x-5}?

Tip: Always remember that the domain of the inverse function is the range of the original function, and vice versa.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Inverse Functions
Natural Logarithm

Formulas

Inverse function formula for exponential functions

Theorems

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Suitable Grade Level

Advanced High School and above