Find a formula for the inverse of the function.
f(x) = e7x − 5
f −1(x)

To find the inverse of the function \( f(x) = e^{7x} - 5 \), follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
\[
y = e^{7x} - 5
\]

### Step 2: Solve for \( x \)
1. Add 5 to both sides:
\[
y + 5 = e^{7x}
\]

2. Take the natural logarithm (ln) of both sides to remove the exponential:
\[
\ln(y + 5) = 7x
\]

3. Solve for \( x \) by dividing both sides by 7:
\[
x = \frac{\ln(y + 5)}{7}
\]

### Step 3: Replace \( y \) with \( x \) to express the inverse function
\[
f^{-1}(x) = \frac{\ln(x + 5)}{7}
\]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{\ln(x + 5)}{7}
\]

Let me know if you'd like more details on any step or clarification!

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Here are five related questions:
1. How do you verify that a function and its inverse are correct?
2. What is the domain of the inverse function \( f^{-1}(x) \)?
3. How does the natural logarithm function work with exponential functions?
4. What happens if you try to find the inverse of \( f(x) = e^{kx} - c \), where \( k \) and \( c \) are constants?
5. How does this process change if the base of the exponential function is not \( e \)?

**Tip:** When finding inverses of functions, always check that the original function is one-to-one (i.e., passes the horizontal line test). This ensures that the function has an inverse.

Find a formula for the inverse of the function f(x) = e^(7x) − 5

Discover how to find the inverse of the exponential function f(x) = e^(7x) - 5 using logarithmic properties. This detailed guide breaks down each step, perfect for high school students studying functions and their inverses in Grades 10-12.

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