Math Problem Statement

For each graph below, state whether it represents a function.

Solution

To determine whether each graph represents a function, we can apply the Vertical Line Test. According to this test, if a vertical line drawn through any part of the graph intersects it at more than one point, the graph does not represent a function. Let's analyze each graph:

  1. Graph 1:

    • A vertical line at x=1x = 1 intersects multiple points.
    • Answer: No, it does not represent a function.
  2. Graph 2:

    • A vertical line drawn through any point on the x-axis intersects only once at each position.
    • Answer: Yes, it represents a function.
  3. Graph 3:

    • A vertical line through the center of the graph intersects it at two points, meaning it fails the test.
    • Answer: No, it does not represent a function.
  4. Graph 4:

    • A vertical line at x=0x = 0 intersects the diamond shape at two points.
    • Answer: No, it does not represent a function.
  5. Graph 5:

    • A vertical line at x=0x = 0 intersects the graph in two separate places.
    • Answer: No, it does not represent a function.
  6. Graph 6:

    • A vertical line intersects each point on the line only once.
    • Answer: Yes, it represents a function.

Summary Table:

GraphFunction?
Graph 1No
Graph 2Yes
Graph 3No
Graph 4No
Graph 5No
Graph 6Yes

Would you like further details or have any questions?

Here are some related questions for further exploration:

  1. How does the vertical line test determine if a graph is a function?
  2. Can a graph fail the vertical line test and still represent a relation?
  3. How does the concept of functions extend to three dimensions?
  4. What are real-world examples of graphs that represent functions?
  5. How does the horizontal line test relate to one-to-one functions?

Tip: Remember that functions must map each input to a unique output; no two points on a vertical line should share the same x-coordinate.

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Math Problem Analysis

Mathematical Concepts

Functions
Vertical Line Test
Graph Analysis

Formulas

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Theorems

Vertical Line Test

Suitable Grade Level

Grades 8-10