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Is this a function and a one-one function? what is the domain and range?

From the image, we can observe a red curve plotted on a coordinate plane. Let's address your questions one by one:

### 1. **Is it a function?**
Yes, the curve shown represents a function. A function passes the **vertical line test**, which states that if any vertical line intersects the graph at more than one point, it is not a function. In this case, every vertical line intersects the curve at only one point, so it satisfies the condition to be a function.

### 2. **Is it a one-to-one function?**
No, it is not a one-to-one function. A one-to-one function passes the **horizontal line test**, which means that any horizontal line should intersect the graph at most once. In the given graph, there are horizontal lines (e.g., around \( y = 5 \)) that intersect the curve at more than one point, so the function is not one-to-one.

### 3. **What is the domain?**
The domain of a function refers to all the possible values of \( x \) (input values) for which the function is defined. From the graph, it appears that the curve extends indefinitely to the left and right, meaning the domain is:
\[
\text{Domain} = (-\infty, \infty)
\]

### 4. **What is the range?**
The range of a function refers to all the possible values of \( y \) (output values). Observing the graph, the lowest point seems to be around \( y = -10 \), and the highest point goes up beyond \( y = 10 \). Therefore, the range is approximately:
\[
\text{Range} = (-\infty, \infty)
\]
since the curve seems to extend vertically in both directions without bound.

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Would you like a more detailed explanation or further clarification?

Here are 5 related questions you might find interesting:
1. How can you algebraically check if a function is one-to-one?
2. What is the significance of a function's domain and range in real-world applications?
3. Can a function be many-to-one and still be considered a function?
4. How does the concept of injectivity relate to one-to-one functions?
5. What are examples of functions that are not one-to-one but are still widely used?

**Tip:** If you're unsure whether a function is one-to-one, always apply both the vertical and horizontal line tests!

Is this a function and a one-one function? What is the domain and range?

This page explores how to determine if a given graph is a function and whether it is one-to-one using the vertical and horizontal line tests. The graph's domain and range are also analyzed in detail.

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