Determine whether each function graphed or defined is one-to-one.

Verified step by step guidance

1. Identify the function type: The graph shown is a parabola, which is a quadratic function.
2. Understand the definition of a one-to-one function: A function is one-to-one if and only if every horizontal line intersects the graph of the function at most once.
3. Apply the Horizontal Line Test: Draw horizontal lines across the graph. If any horizontal line intersects the graph more than once, the function is not one-to-one.
4. Analyze the intersections: Observe that horizontal lines intersect the parabola at two points in most cases.
5. Conclude: Since horizontal lines intersect the graph at more than one point, the function is not one-to-one.

Key Concepts

One-to-One Function

A function is considered one-to-one if each output value corresponds to exactly one input value. This means that no two different inputs can produce the same output. To determine if a function is one-to-one, one can use the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

Horizontal Line Test

The horizontal line test is a visual method used to determine if a function is one-to-one. If a horizontal line drawn across the graph intersects the curve at more than one point, the function fails the test and is not one-to-one. This test is particularly useful for analyzing the graphs of functions, such as parabolas, to quickly assess their one-to-one nature.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, which can open upwards or downwards. Since parabolas are symmetric, they often fail the horizontal line test, indicating that they are not one-to-one functions.