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The question is asking if the function \( f(x) = x^2 + 2 \) is one-to-one. A function is considered one-to-one if every unique input maps to a unique output, meaning no two different values of \( x \) produce the same \( f(x) \).

The function \( f(x) = x^2 + 2 \) represents a parabola that opens upwards, with its vertex at \( (0, 2) \). For a function to be one-to-one, it must pass the **horizontal line test**, where no horizontal line intersects the graph at more than one point. However, in this case, horizontal lines will intersect the graph of \( f(x) = x^2 + 2 \) at two points for values of \( f(x) > 2 \), which means it fails the horizontal line test.

**Conclusion**: This function is **not one-to-one**.

Would you like more details or have any questions on this?

Here are some additional questions related to this topic:
1. What does it mean for a function to be one-to-one in terms of injectivity?
2. How can we determine if a function is one-to-one just by looking at its equation?
3. How does the horizontal line test apply to other functions, like linear functions?
4. Can we restrict the domain of \( f(x) = x^2 + 2 \) to make it one-to-one?
5. How does the concept of an inverse function relate to a function being one-to-one?

**Tip**: To determine if a function is one-to-one, try applying the horizontal line test on its graph or check if every input value leads to a unique output.

Is this function one-to-one? f(x) = x^2 + 2

Understand if the quadratic function f(x) = x^2 + 2 is one-to-one by applying the horizontal line test. Explore key concepts in algebra and quadratic functions, suitable for students in Grades 9-11.

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