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find the inverse of this graph on a table 

To find the inverse of the graph from the image, we need to create a table for the inverse function. An inverse of a function \( f(x) \) essentially swaps the \( x \)- and \( y \)-coordinates of points on the original graph. Let's identify some key points from the graph and then switch their coordinates.

### Step 1: Identifying Points on the Graph

Based on the graph, we can estimate a few points:

- \( f(0) = 0 \)
- \( f(1) \approx 1 \)
- \( f(5) \approx 3 \)
- \( f(10) \approx 5 \)
- \( f(15) \approx 7 \)

### Step 2: Creating the Table for \( f^{-1}(x) \)

To find the inverse, we swap the \( x \)- and \( y \)-coordinates of these points:

| \( f(x) \) (Original Function) | \( f^{-1}(x) \) (Inverse Function) |
|------------------------------|------------------------------------|
| \( (0, 0) \)                 | \( (0, 0) \)                      |
| \( (1, 1) \)                 | \( (1, 1) \)                      |
| \( (5, 3) \)                 | \( (3, 5) \)                      |
| \( (10, 5) \)                | \( (5, 10) \)                     |
| \( (15, 7) \)                | \( (7, 15) \)                     |

### Step 3: Interpretation

The table shows the points for the inverse function \( f^{-1}(x) \), where the roles of \( x \) and \( y \) are reversed.

Would you like further clarification on how the inverse is constructed, or do you want to explore specific properties of the function?

### Related Questions:
1. How can you verify if two functions are inverses algebraically?
2. What are the key features of the inverse function from the graph?
3. What does it mean for a function to have an inverse?
4. How does the domain and range change for the inverse of a function?
5. Can every function have an inverse? If not, why?

**Tip**: To find the inverse graphically, you can reflect the original graph across the line \( y = x \).

Find the inverse of this graph on a table.

This article provides a detailed guide on finding the inverse of a function based on its graph. Learn how to create a table of values for the inverse by swapping the x- and y-coordinates from the graph. Suitable for high school students (Grades 9-12).

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