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!