https://r2.erweima.ai/mathbot/image/1725745966826weaf25vu.jpg

The statement in the image reads: "La función representada por la gráfica no tiene inversa," which translates to "The function represented by the graph does not have an inverse."

To check if this statement is true or false, we need to analyze the graph. For a function to have an inverse, it must be **bijective**, which means it must be both:

1. **Injective (one-to-one)**: No two different \( x \)-values should map to the same \( y \)-value. This is typically checked with the horizontal line test: if any horizontal line intersects the graph more than once, the function is not injective and therefore does not have an inverse.
   
2. **Surjective (onto)**: The function must cover the entire range of possible \( y \)-values.

In this graph, the curve appears to fail the **horizontal line test**. There are portions of the graph where a horizontal line could intersect the curve more than once, indicating that the function is **not injective**. Therefore, the function **does not have an inverse**, and the statement is **true**.

Would you like more details, or do you have any questions about this?

Here are 5 related questions to consider:
1. What is the horizontal line test, and how is it applied to graphs?
2. What are the necessary conditions for a function to have an inverse?
3. How can a function be restricted to make it invertible?
4. What is the difference between injective and surjective functions?
5. How can you algebraically determine if a function has an inverse?

**Tip:** Always check whether a function is injective by using the horizontal line test on its graph to ensure it is one-to-one.

Analyze whether a function represented by a graph has an inverse using the horizontal line test. Understand key mathematical concepts like injectivity and surjectivity and learn how to determine the existence of an inverse function. Suitable for students in Grades 9-12.

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