Math Problem Statement

Use the Vertical Line Test to determine whether the curve is the graph of a function of x. If the curve is a function, state the domain and range.

Solution

Let's analyze the graph you provided:

  1. Vertical Line Test: This test helps determine whether a curve is a function of xx. If any vertical line intersects the curve at more than one point, the curve is not a function of xx. Upon inspecting the graph, it appears that no vertical line would intersect the curve more than once at any point. Therefore, the curve is a function of xx.

  2. Domain: The domain of a function is the set of all possible xx-values for which the function is defined. Looking at the graph, the xx-values range from 4-4 to 44, so the domain in interval notation is: Domain=[4,4]\text{Domain} = [-4, 4]

  3. Range: The range is the set of all possible yy-values the function can take. From the graph, the yy-values range from 4-4 to 44, so the range in interval notation is: Range=[4,4]\text{Range} = [-4, 4]

Summary:

  • The curve is a function of xx.
  • Domain: [4,4][-4, 4]
  • Range: [4,4][-4, 4]

Would you like any further explanation or details?

Here are 5 related questions for deeper understanding:

  1. What are some other methods for determining if a curve is a function besides the vertical line test?
  2. How would the domain and range change if the curve extended beyond x=4x = 4 or y=4y = 4?
  3. Can a function ever have more than one yy-value for a single xx-value?
  4. How would you find the inverse of a function, and what would be the domain and range of the inverse?
  5. Why is it important to express the domain and range in interval notation?

Tip: When analyzing the domain and range, always pay attention to the extreme points of the graph to capture the full scope of possible values.

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Math Problem Analysis

Mathematical Concepts

Functions
Vertical Line Test
Domain
Range

Formulas

None - Concept-based analysis

Theorems

Vertical Line Test

Suitable Grade Level

Grades 9-12