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.

Explore how to identify a one-to-one (injective) function using the horizontal line test. This explanation includes graph analysis, examples like f(x) = x^3 and f(x) = x^2, and covers the key concept of one-to-one functions. Ideal for high school students in Grades 10-12.

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