A one-to-one function (also called an injective function) is a function where each input is mapped to a unique output. In other words, no two different inputs will produce the same output. This property makes the function invertible, meaning you can reverse the function to find the input from the output.

Graph of a One-to-One Function

To determine if a function is one-to-one from its graph, it must pass the Horizontal Line Test:

• Horizontal Line Test: If no horizontal line intersects the graph of the function more than once, the function is one-to-one. This is because a horizontal line checks if different inputs (along the x-axis) lead to the same output (along the y-axis). If the line intersects the graph in more than one point, the function is not one-to-one.

Examples:

• The graph of $f(x) = x^3$ is one-to-one because it passes the horizontal line test.
• The graph of $f(x) = x^2$ is not one-to-one because horizontal lines (like $y = 1$) intersect the graph at two points (e.g., $x = 1$ and $x = -1$).

Would you like more details or have any specific questions?

Related Questions:

1. What is the difference between one-to-one and onto functions?
2. How do you find the inverse of a one-to-one function?
3. What types of functions are always one-to-one?
4. Can a function be one-to-one but not continuous?
5. How does the vertical line test relate to the horizontal line test?

Tip: When graphing, the shape of the curve can often give hints as to whether or not a function is one-to-one.