Functions

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.

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To determine if a graph represents a function of x, we can use the 'Vertical Line Test'. This test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.

Examine the given graph visually or conceptually. Imagine drawing vertical lines (lines parallel to the y-axis) across the entire graph.

For each vertical line you imagine, check if it intersects the graph at more than one point. If any vertical line does intersect the graph at more than one point, then the graph is not a function of x.

If every vertical line intersects the graph at most once, then the graph is a function of x.

Provide a reason based on the Vertical Line Test: If the graph passes the test, state that it is a function of x because no vertical line intersects it more than once. If it fails, state that it is not a function of x because there exists at least one vertical line that intersects it more than once.

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Definition of a Function

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. This means that for every x-value in the domain, there is a unique y-value in the range. Understanding this definition is crucial for determining whether a graph represents a function.

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Definition of the Definite Integral

Vertical Line Test

The vertical line test is a method used to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the graph does not represent a function. This test provides a visual way to assess the uniqueness of outputs for given inputs.

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Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). Understanding the domain and range helps in analyzing the behavior of functions and identifying whether a graph meets the criteria of a function based on its visual representation.

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