To find the inverse of the function \( f(x) = e^{5x} - 1 \), follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
We start by writing the function as:
\[
y = e^{5x} - 1
\]

### Step 2: Solve for \( x \)
Now, we need to isolate \( x \) in terms of \( y \). First, add 1 to both sides:
\[
y + 1 = e^{5x}
\]
Next, take the natural logarithm (ln) of both sides to remove the exponential:
\[
\ln(y + 1) = 5x
\]
Now, solve for \( x \) by dividing both sides by 5:
\[
x = \frac{1}{5} \ln(y + 1)
\]

### Step 3: Replace \( y \) with \( x \)
Since we're finding the inverse, we replace \( y \) with \( x \), giving the inverse function:
\[
f^{-1}(x) = \frac{1}{5} \ln(x + 1)
\]

Thus, the inverse of the function \( f(x) = e^{5x} - 1 \) is:
\[
f^{-1}(x) = \frac{1}{5} \ln(x + 1)
\]

Do you want more details or have any questions? Here are 5 related questions for practice:

1. What is the domain of the inverse function \( f^{-1}(x) \)?
2. How do you find the inverse of a logarithmic function?
3. What is the range of \( f(x) = e^{5x} - 1 \)?
4. Can you find the derivative of \( f^{-1}(x) \)?
5. What happens to the graph of a function when you take its inverse?

**Tip:** Always check that the function is one-to-one before finding its inverse, as not all functions have inverses.

Learn how to find the inverse of the exponential function f(x) = e^{5x} - 1 in this step-by-step guide. This solution covers key mathematical concepts, including inverse functions, logarithms, and properties of exponential functions. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation