Determining if a Function is One-to-One from Its Graph

Q6. Decide if each function graphed is 1-to-1.

Background

Topic: One-to-One Functions (Injective Functions)

This question is testing your ability to determine, from a graph, whether a function is one-to-one. A function is one-to-one if every horizontal line intersects the graph at most once. This is known as the Horizontal Line Test.

Key Terms and Concepts:

• One-to-One Function: A function $f$ is one-to-one if $f(a) = f(b)$ implies $a = b$ for all $a, b$ in the domain.

• Horizontal Line Test: If every horizontal line crosses the graph of the function at most once, the function is one-to-one.

Step-by-Step Guidance

1. Examine each graph and imagine drawing horizontal lines (parallel to the x-axis) across the entire graph.

2. For each graph, count how many times a horizontal line could intersect the graph. If any horizontal line crosses the graph more than once, the function is not one-to-one.

3. For the top row of graphs (first image):

   • Top graph: Does any horizontal line intersect the graph more than once?

   • Bottom graph: Repeat the horizontal line test.

   

4. For the middle row of graphs (second image):

   • Top graph: Apply the horizontal line test.

   • Bottom graph: Apply the horizontal line test.

   

5. For the bottom row of graphs (third image):

   • Top graph: Apply the horizontal line test.

   • Bottom graph: Apply the horizontal line test.

   

Try solving on your own before revealing the answer!

Final Answer:

Top row: No, Yes, No, Yes Bottom row: No, No, No, No

Only the graphs that pass the horizontal line test are one-to-one.