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

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Step 1: Identify the relation given in the graph. The graph shows a parabola opening downwards with a vertex at the point (8, -2).

Step 2: Determine if the relation is a function. A relation is a function if each input (x-value) corresponds to exactly one output (y-value). Use the vertical line test: if any vertical line intersects the graph at more than one point, it is not a function.

Step 3: Apply the vertical line test to the graph. Notice that any vertical line drawn through the graph intersects it at most once, confirming that the relation is a function.

Step 4: Determine the domain of the function. The domain is the set of all possible x-values. Since the parabola extends infinitely to the left and right, the domain is all real numbers.

Step 5: Determine the range of the function. The range is the set of all possible y-values. Since the parabola opens downwards and the highest point is at y = -2, the range is all real numbers less than or equal to -2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Definition

A function is a relation where each input (or domain value) corresponds to exactly one output (or range value). This means that for any x-value in the domain, there can only be one y-value. To determine if a relation is a function, one can use the vertical line test: if a vertical line intersects the graph at more than one point, the relation is not a function.

Domain and Range

The domain of a function is the complete set of possible input values (x-values) that the function can accept, while the range is the complete set of possible output values (y-values) that the function can produce. For example, in a quadratic function, the domain is typically all real numbers, but the range is limited to values above or below the vertex, depending on the direction of the parabola.

Quadratic Functions

A quadratic function is a type of polynomial function represented by the equation 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. The vertex of the parabola represents the maximum or minimum point of the function, and understanding its position is crucial for determining the range.

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