For each graph, determine whether y is a function of x. Give the domain and range of each relation.

검증된 단계별 안내

Step 1: Identify if y is a function of x by using the vertical line test. A relation is a function if and only if no vertical line intersects the graph at more than one point.

Step 2: Observe the graph. Notice that for some values of x, there are two corresponding y-values (for example, at x = 0, y = 11 and y = -11). This means the vertical line at x = 0 intersects the graph at two points.

Step 3: Since the vertical line at x = 0 intersects the graph at two points, y is not a function of x for this relation.

Step 4: Determine the domain by looking at the x-values covered by the graph. The graph extends horizontally from about -20 to 20, so the domain is approximately \([-20, 20]\).

Step 5: Determine the range by looking at the y-values covered by the graph. The graph extends vertically from about -20 to 20, so the range is approximately \([-20, 20]\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 at more than one point. Understanding this helps determine if y is a function of x by checking if any x-value maps to multiple y-values.

Domain and Range

The domain of a relation is the set of all possible x-values, while the range is the set of all possible y-values. Identifying these sets from a graph involves looking at the horizontal spread for the domain and the vertical spread for the range.

Vertical Line Test

The vertical line test is a visual method to determine if a graph represents a function. If any vertical line drawn through the graph intersects it more than once, the graph does not represent a function. This test is essential for analyzing the given parabolic curves.