https://file.aiquickdraw.com/mathbot/file/1726886376416i1sgvbrw.jpg

The problem asks to find the inverse of the function \( f(x) = 7x - 9 \), and also mentions that the graphs of \( f \) and \( f^{-1} \) are symmetric with respect to the line \( y = \) (a specific value needs to be filled in).

### Part (a): Finding the inverse function
Given \( f(x) = 7x - 9 \), to find the inverse, follow these steps:
1. Replace \( f(x) \) with \( y \), so the equation becomes:
   \[
   y = 7x - 9
   \]
2. Swap \( x \) and \( y \) to begin finding the inverse:
   \[
   x = 7y - 9
   \]
3. Solve for \( y \):
   \[
   x + 9 = 7y
   \]
   \[
   y = \frac{x + 9}{7}
   \]
4. Therefore, the inverse function is:
   \[
   f^{-1}(x) = \frac{x + 9}{7}
   \]

### Part (b): Symmetry of graphs
The graphs of a function and its inverse are always symmetric with respect to the line \( y = x \). Therefore, the answer for this part is:
\[
y = x
\]

---

To summarize:
- Part (a): \( f^{-1}(x) = \frac{x + 9}{7} \)
- Part (b): \( y = x \)

Do you need any further details or clarifications?

Here are five related questions to help deepen your understanding:
1. How do you verify if two functions are inverses of each other?
2. Why are the graphs of a function and its inverse symmetric with respect to \( y = x \)?
3. Can every function have an inverse? What conditions are necessary for a function to be invertible?
4. How would you graphically represent both \( f(x) = 7x - 9 \) and \( f^{-1}(x) \) on the same coordinate plane?
5. How does the derivative of a function relate to the derivative of its inverse?

**Tip**: When finding the inverse of a function, swapping \( x \) and \( y \) is key, but don’t forget to check if the original function is one-to-one to ensure it has an inverse!

Find the inverse function of f(x) = 7x - 9 and identify the line of symmetry between f and its inverse.

Learn how to find the inverse of the function f(x) = 7x - 9 and explore the symmetry between f and f^(-1) with respect to the line y = x. This problem involves key algebraic concepts and theorems, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation