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Using the horizontal-line test, determine whether the function is one-to-one.

The graph provided appears to represent a portion of a function, and you are asking to determine if it is one-to-one using the **horizontal line test**.

### **Horizontal Line Test**
A function is **one-to-one** if and only if **any horizontal line** intersects the graph at most **once**. This means that each value of \( y \) should correspond to only one value of \( x \), indicating that the function has no repeating \( y \)-values for different \( x \)-values.

### Analysis of the Graph
The graph shows a curve resembling a **portion of a rational or exponential function** that decreases rapidly in the first quadrant and extends into the second quadrant (as \( x \to 0 \) from the left and \( y \to +\infty \)). 

If we draw **horizontal lines** across the graph, each horizontal line intersects the curve **only once** for each \( y \)-value. This confirms that no \( y \)-value corresponds to more than one \( x \)-value.

### Conclusion
Since no horizontal line intersects the graph more than once, the function passes the horizontal-line test, and **it is a one-to-one function**.

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Would you like further details or have any specific questions about this?  
Here are some related questions to explore this concept further:

1. What kind of functions generally pass the horizontal line test?
2. How does the horizontal line test relate to the function's inverse?
3. Can a polynomial function be one-to-one?
4. What are examples of functions that fail the horizontal-line test?
5. How does the graph of a one-to-one function differ from a non-one-to-one function?

**Tip:** Inverse functions exist only for one-to-one functions, so passing the horizontal-line test ensures that the function has an inverse!

Learn how to use the horizontal-line test to determine if a function is one-to-one. This analysis involves understanding the graphical behavior of functions and their ability to pass the test, ensuring that each y-value corresponds to only one x-value. Suitable for high school students studying functions.

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