College Algebra: One-to-One Functions and Inverses (Graphical Analysis)

Q1. Determine whether or not the function shown in the graph is one-to-one.

Background

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

This question is testing your ability to determine if a function is one-to-one by analyzing its graph. A function is one-to-one if every horizontal line intersects the graph at most once.

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 $a$ and $b$ in the domain.

• Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more than once.

Step-by-Step Guidance

1. Observe the shape of the graph. Identify if it is a straight line, parabola, or another type of curve.

2. Recall the horizontal line test: Imagine drawing horizontal lines ($y = c$ for various values of $c$) across the graph.

3. Check if any horizontal line would intersect the graph at more than one point. If so, the function is not one-to-one.

4. Think about the definition: If two different $x$-values can produce the same $y$-value, the function fails to be one-to-one.

Try solving on your own before revealing the answer!

Final Answer:

The function is not one-to-one. This is because a horizontal line can intersect the parabola at two points, meaning two different $x$-values can yield the same $y$-value.

This violates the definition of a one-to-one function.