Math Problem Statement

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, ∞)

Solution

The quadratic parent function f(x)=x2f(x) = x^2 is not inherently one-to-one because it fails the horizontal line test; for every yy-value, there can be two corresponding xx-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 xx-values, ensuring that the function passes the horizontal line test because every output yy corresponds to exactly one input xx.

Why the other options don't work:

  1. D: (−∞, 0) ∪ (0, ∞) — This still includes two different xx-values that can yield the same yy-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 xx-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,)[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).

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Functions
Quadratic Parent Function
One-to-One Functions

Formulas

Quadratic Function f(x) = x^2

Theorems

Horizontal Line Test

Suitable Grade Level

Grades 9-12