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From these 4 graphs, which of the following graphs could be showing a function and its inverse function

To determine which of the graphs represent a function and its inverse, we need to remember that a function and its inverse will reflect over the line \( y = x \). Additionally, each must pass the vertical line test to be considered a function.

Let's evaluate each graph based on this:

1. **First Graph (Top-left)**: The red and blue curves appear to be reflections of each other across the line \( y = x \). The red curve is an exponential-like curve, and the blue curve seems to be its inverse (a logarithmic-like curve). Both pass the vertical line test, so this could represent a function and its inverse.

2. **Second Graph (Top-right)**: The red curve shows a more complex behavior, with a portion of the graph not reflecting a typical one-to-one function. The blue curve also doesn't seem to be a clear inverse, as the reflection over \( y = x \) doesn't hold throughout the graph. This one does **not** seem to show a function and its inverse.

3. **Third Graph (Bottom-left)**: This graph shows a parabola (red) and what appears to be a sideways parabola (blue). A parabola does not have an inverse that is also a function unless it is restricted. The sideways parabola (blue) does not pass the vertical line test, meaning it is not a valid function. Thus, this graph does **not** show a function and its inverse.

4. **Fourth Graph (Bottom-right)**: The red and blue lines appear to be reflections across the line \( y = x \). Both are linear, and linear functions are always one-to-one and have inverses. Thus, this graph **does** represent a function and its inverse.

### Conclusion:
The graphs that show a function and its inverse are:
- **First graph (top-left)**
- **Fourth graph (bottom-right)**

Would you like more details or further explanation?

### Related Questions:
1. What are the conditions for a function to have an inverse?
2. How can you test if two graphs are inverses of each other?
3. Why doesn't a parabola have an inverse unless restricted?
4. What is the significance of the line \( y = x \) in relation to inverse functions?
5. Can a function and its inverse intersect?

### Tip:
To quickly check if two functions are inverses graphically, see if they reflect over the line \( y = x \) and each still passes the vertical line test!

Identify which of the following graphs could be showing a function and its inverse function.

Learn how to identify which graphs represent a function and its inverse by checking reflections over y = x and applying the vertical line test. This problem covers key algebraic concepts related to inverse functions, reflection properties, and one-to-one functions.

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