Precalculus Review: One-to-One Functions and Graphs

Q15. Determine whether the function is one-to-one. Explain.

Background

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

This question is testing your understanding of what it means for a function to be one-to-one. A function is one-to-one if every y-value in the range is paired with exactly one x-value in the domain. In other words, no horizontal line should intersect the graph of the function more than once (Horizontal Line Test).

Key Terms and Concepts:

• One-to-One Function: A function $f(x)$ 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 intersects the graph of the function at most once, the function is one-to-one.

Step-by-Step Guidance

1. Examine the graph provided. Notice the shape and orientation of the curve.

   

2. Apply the horizontal line test: Imagine drawing horizontal lines at various y-values. Observe whether any horizontal line crosses the graph more than once.

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

4. Think about the algebraic form of the function that matches the graph (for example, $y = |x|$ or $y = x^2$) and recall whether these are one-to-one.

Try solving on your own before revealing the answer!

Final Answer:

The function shown is not one-to-one because there exist horizontal lines that intersect the graph at more than one point. For example, the line $y = 1$ crosses the graph at two points.

This means there are different x-values that produce the same y-value, so the function fails the horizontal line test.