Math Problem Statement

f^-1(x) = e^2x+5

Solution

The given function is:

f1(x)=e2x+5f^{-1}(x) = e^{2x + 5}

This represents the inverse function of some original function f(x)f(x). To find f(x)f(x), we need to follow these steps:

Step 1: Replace f1(x)f^{-1}(x) with yy

Let y=f1(x)=e2x+5y = f^{-1}(x) = e^{2x + 5}.

Step 2: Swap xx and yy

We switch xx and yy to find the original function: x=e2y+5.x = e^{2y + 5}.

Step 3: Solve for yy

To solve for yy, follow these steps:

  1. Take the natural logarithm on both sides to eliminate the exponential function: ln(x)=2y+5.\ln(x) = 2y + 5.
  2. Isolate yy: 2y=ln(x)5,2y = \ln(x) - 5, y=ln(x)52.y = \frac{\ln(x) - 5}{2}.

Thus, the original function is: f(x)=ln(x)52.f(x) = \frac{\ln(x) - 5}{2}.

Summary:

The original function is: f(x)=ln(x)52.f(x) = \frac{\ln(x) - 5}{2}.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

  1. How do you find the derivative of f(x)=ln(x)52f(x) = \frac{\ln(x) - 5}{2}?
  2. What are the domain and range of f(x)f(x)?
  3. How do you compute the inverse of logarithmic functions in general?
  4. How can we verify that f(f1(x))=xf(f^{-1}(x)) = x and f1(f(x))=xf^{-1}(f(x)) = x?
  5. How does the natural logarithm function relate to exponential functions?

Tip: When working with logarithmic and exponential equations, taking logarithms on both sides is a common strategy to simplify the expression.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Exponential Functions
Logarithmic Functions

Formulas

Inverse of an exponential function
Natural logarithm: ln(x)
Basic algebraic manipulation

Theorems

Inverse Function Theorem
Properties of Logarithms and Exponentials

Suitable Grade Level

Grades 10-12