Which of the following is a domain restriction on the quadratic parent function that makes it
one-to-one?


Which of the following is a domain restriction on the quadratic parent function that makes it
one-to-one?


D: (−∞, 0) ∪ (0, ∞)


There is no domain restriction needed.


D: {x|x < 1}


D: [0, ∞)

The quadratic parent function \( f(x) = x^2 \) is not inherently one-to-one because it fails the horizontal line test; for every \( y \)-value, there can be two corresponding \( x \)-values. However, it can be made one-to-one by restricting its domain.

Out of the given options, the correct domain restriction that makes the quadratic parent function one-to-one is:

**D: [0, ∞)**

This restricts the domain to non-negative \( x \)-values, ensuring that the function passes the horizontal line test because every output \( y \) corresponds to exactly one input \( x \).

### Why the other options don't work:
1. **D: (−∞, 0) ∪ (0, ∞)** — This still includes two different \( x \)-values that can yield the same \( y \)-value, so it does not make the function one-to-one.
2. **There is no domain restriction needed.** — This is incorrect because, without a restriction, the quadratic function is not one-to-one.
3. **D: {x|x < 1}** — While this restricts the domain, it still allows negative \( x \)-values, which cause the function to fail the horizontal line test.

Would you like more details or have any further questions?

Here are 5 related questions for further exploration:
1. What is the definition of a one-to-one function?
2. How do you apply the horizontal line test to check if a function is one-to-one?
3. What is the range of the quadratic function when its domain is restricted to \( [0, ∞) \)?
4. How can the inverse of a quadratic function be found if the domain is restricted?
5. What transformations can affect whether a function is one-to-one?

**Tip:** When restricting domains, focus on isolating sections where the function is monotonic (strictly increasing or decreasing).

Which of the following is a domain restriction on the quadratic parent function that makes it one-to-one?
D: (−∞, 0) ∪ (0, ∞)
There is no domain restriction needed.
D: {x|x < 1}
D: [0, ∞)

Discover how to make the quadratic parent function one-to-one by applying the correct domain restrictions. Learn about the horizontal line test, quadratic functions, and how domain affects function properties. Suitable for high school students in Grades 9-12.

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