Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

Verified step by step guidance

1. Step 1: Determine if the relation is a function by using the vertical line test. If any vertical line intersects the graph at more than one point, the relation is not a function. For this graph, each vertical line intersects the parabola at exactly one point, so it defines a function.
2. Step 2: Identify the domain of the function. The domain consists of all possible x-values for which the function is defined. Since the parabola extends infinitely to the left and right, the domain is all real numbers, which can be written as \((-\infty, \infty)\).
3. Step 3: Identify the range of the function. The range consists of all possible y-values that the function can take. The vertex of the parabola is at (8, -2), and since the parabola opens downward, the maximum y-value is -2.
4. Step 4: Express the range using interval notation. Since the parabola opens downward and extends infinitely downward, the range includes all y-values less than or equal to -2, which is \((-\infty, -2]\).
5. Step 5: Summarize the findings: The relation is a function, the domain is all real numbers \((-\infty, \infty)\), and the range is \((-\infty, -2]\).

Key Concepts

Definition of a Function

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). This means no vertical line intersects the graph more than once, ensuring each x has a unique y. The vertical line test is a common method to verify if a graph represents a function.

Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined. For the given quadratic graph, the domain includes all x-values where the graph exists, typically all real numbers unless restricted by the graph's endpoints or breaks.

Range of a Function

The range is the set of all possible output values (y-values) that the function can produce. For the parabola shown, the range includes all y-values from the vertex's minimum or maximum point (here, the vertex at (8, -2)) extending upwards or downwards depending on the parabola's direction.