Math Problem Statement

WRONG: Answer Explanation Correct answer: [ − 2 , 2 ]

[ − 6 , − 2 ]

A function will be one-to-one if it passes the horizontal line test. We can make the function shown pass the horizontal line test by taking either half of the graph (where it is increasing, or where it is decreasing).

So we can either take the left half, which is the interval
[ − 6 , − 2 ] , or the right half, which is the interval
[ − 2 , 2 ] .

Note: we could also take our domain to be an interval which is contained in the left or right half, for instance
[ − 5 , − 3 ] or
[ − 1 , 1 ] , but typically we want to choose a domain which has all the values of
f ( x ) as its corresponding range.

Your answer: [ − 6 , − 2 ]

[ − 5 , − 1 ]

Solution

Thank you for the clarification. Let's reassess the answer with this understanding.

To determine the intervals where the function is one-to-one, we must find intervals where the function is either strictly increasing or strictly decreasing.

The graph indicates that the function is strictly increasing on the interval [−6,−2][-6, -2] and strictly decreasing on the interval [−2,2][-2, 2]. This means:

  • [−6,−2][-6, -2]: The function is strictly increasing in this interval.
  • [−2,2][-2, 2]: The function is strictly decreasing in this interval.

Thus, the correct intervals where the function is one-to-one are:

  • [−6,−2][-6, -2]
  • [−2,2][-2, 2]

Would you like more details or have any questions?

Here are 5 more questions you might consider asking:

  1. How can you visually determine if a function is strictly increasing or decreasing on an interval?
  2. What is the Horizontal Line Test, and how does it relate to one-to-one functions?
  3. How do you determine the inverse of a piecewise function?
  4. Can you give examples of real-world situations modeled by one-to-one functions?
  5. What are the implications of a function being one-to-one for its domain and range?

Tip: When solving problems involving the determination of one-to-one intervals, always consider if the function is strictly increasing or strictly decreasing in the proposed interval.

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Math Problem Analysis

Mathematical Concepts

Functions
One-to-One Functions
Horizontal Line Test

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12

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