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 across different $y$-values.

2. For each graph, check if any horizontal line would intersect the graph at more than one point.

3. If a horizontal line crosses the graph more than once, the function is not one-to-one.

4. If every horizontal line crosses the graph at most once, the function is one-to-one.

5. Repeat this process for each graph provided.

Try solving on your own before revealing the answer!

Final Answer:

Top row: not one-to-one, one-to-one, not one-to-one, one-to-one. Bottom row: not one-to-one, not one-to-one, not one-to-one, not one-to-one.

Only the graphs that pass the horizontal line test are one-to-one. Most common functions like parabolas and absolute value functions are not one-to-one, while lines with nonzero slope and some cubic or exponential functions can be one-to-one.