https://file.aiquickdraw.com/mathbot/file/1726109241341q00c8rwu.jpg

The image asks you to determine which of the provided items represent functions. Let's analyze each one.

1. **Set of ordered pairs:**
   \[
   \{(-2, 4), (1, 5), (3, 4), (3, 5)\}
   \]
   This is a relation, and to determine if it's a function, we need to check if each input (x-value) maps to exactly one output (y-value). Notice that the input \( x = 3 \) maps to both 4 and 5, which violates the definition of a function. **This is not a function.**

2. **Equation:**
   \[
   y^3 = 2x^2 - 7
   \]
   This equation involves powers of both \( y \) and \( x \), and to determine if it defines a function, we need to check if for every \( x \)-value, there is only one \( y \)-value. Since \( y^3 \) is involved, each \( x \)-value might correspond to multiple \( y \)-values, meaning this is not a function. **This is not a function.**

3. **Graph of an oval shape:**
   This graph represents an ellipse, which does not pass the vertical line test. The vertical line test states that if a vertical line intersects the graph at more than one point, it is not a function. Since an ellipse will intersect vertical lines at two points, **this is not a function**.

4. **Scenario:**
   "The average miles to the gallon for a car at different speeds." 
   If each speed corresponds to exactly one value for the average miles per gallon, this represents a function. Typically, this scenario describes a relationship where each speed has one corresponding fuel efficiency, making it a function. **This is a function.**

5. **Equation:**
   \[
   x^2 + y^2 = 9
   \]
   This equation represents a circle with a radius of 3, centered at the origin. A circle does not pass the vertical line test, so **this is not a function**.

**Conclusion:**
The only item that represents a function is the scenario describing the average miles per gallon at different speeds.

Let me know if you want further details or have any questions. Here are 5 related questions for further study:

1. How do you apply the vertical line test to determine if a graph represents a function?
2. Why does the equation \( y^3 = 2x^2 - 7 \) fail to represent a function?
3. What defines a relation versus a function in mathematics?
4. Can a circle ever represent a function? Why or why not?
5. How does the concept of one-to-one functions differ from general functions?

**Tip:** Always check whether each input value (x-value) maps to a single output value (y-value) when determining if a relation is a function.

Determine which of the following represents functions: 
1. {(-2, 4), (1, 5), (3, 4), (3, 5)} 
2. y^3 = 2x^2 - 7 
3. Graph of an oval shape 
4. The average miles to the gallon for a car at different speeds 
5. x^2 + y^2 = 9

Learn how to determine which relations and equations represent functions. This question involves analyzing ordered pairs, an algebraic equation, and a graph to check if they meet the definition of a function using methods like the vertical line test.

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