Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

A table for the relation $f$ is below.

$x$	1	2	3	1	4	5
$f (x)$	-1	5	-2	1	3	-2

1. Is this relation a function? Explain your answer.

Compare your answer:

$f$ is not a function because $f (1) = - 1$ and $f (1) = 1$ .

2. Use the Desmos graphing tool or technology outside the course. Graph the points in the relation.

Interact with the graph. Graph the points in the relation on a coordinate grid.

Access multimedia content

Compare your answer:

A scatter plot with five black dots clustered in the lower left quadrant, positioned near the origin, with the t-axis labeled in intervals of 5 from negative 5 to 15 and the x-axis from 0 to 15.

3. What do you notice about the points $(1, - 1)$ and $(1, 1)$ ?

Compare your answer:

They are on the same vertical line.

VERTICAL LINE TEST

A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.

If any vertical line intersects the graph in more than one point, the graph does not represent a function.

If the graph does represent a function, we say it “passes the vertical line test.”

4. For the function graphed below, determine if the graph passes the vertical line test and then explain how this identifies the graph as a function or not.

A line with a negative slope crosses the y-axis at 2 and the x-axis at 3 on a coordinate grid ranging from negative 6 to 6 on both axes. The arrow on the left indicates the line continues beyond the grid.

Compare your answer:

Yes, this passes the vertical line test (see the sample blue lines below) for all points on the line so the graph is a function.

The graph of a straight line that passes through the points (0, 2) and (3, 0), shown in red and blue vertical line passing through (negative 2, 0) a green vertical line passing through (4, 0).

5. For the function graphed below, determine if the graph passes the vertical line test and then explain how this identifies the graph as a function or not.

A sideways parabola that opens to the right and passes through the points (negative 1, 0), (3, 2), and (3, negative 2), a black vertical line passing through (1, 0).

Compare your answer:

No, this graph does not pass the vertical line test, so it is not a function. For example, the vertical line $x = 1$ crosses the graph more than once.

Notice that the graph of a non-vertical line will always be a function. This is called a linear function.

Video: Interpreting Function Notation

Watch the following video to learn more about function notation.

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Self Check

Which of the following graphs is the graph of a function?

Additional Resources

Determine If a Graph Is a Function

This figure has a graph next to a table. The graph has a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The line is labeled f of x equals 2 x minus 3. There are several vertical arrows that relate values on the x-axis to points on the line. The first arrow relates x equals negative 2 on the x-axis to the point (negative 2, negative 7) on the line. The second arrow relates x equals negative 1 on the x-axis to the point (negative 1, negative 5) on the line. The next arrow relates x equals 0 on the x-axis to the point (0, negative 3) on the line. The next arrow relates x equals 3 on the x-axis to the point (3, 3) on the line. The last arrow relates x equals 4 on the x-axis to the point (4, 5) on the line. The table has 7 rows and 3 columns. The first row is a title row with the label f of x equals 2 x minus 3. The second row is a header row with the headers x, f of x, and (x, f of x). The third row has the coordinates negative 2, negative 7, and (negative 2, negative 7). The fourth row has the coordinates negative 1, negative 5, and (negative 1, negative 5). The fifth row has the coordinates 0, negative 3, and (0, negative 3). The sixth row has the coordinates 3, 3, and (3, 3). The seventh row has the coordinates 4, 5, and (4, 5).

A relation is a function if every input has exactly one output value. So the relation defined by the equation $y = 2 x - 3$ is a function. Notice that a line will always be a function, so it is called a linear function.

If we look at the graph, each vertical dashed line intersects the line at only one point. This makes sense because in a function, for every $x$ -value there is only one $y$ -value.

If the vertical line hit the graph twice, the $x$ -value would be mapped to two $y$ -values, so the graph would not represent a function.

This leads us to the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function.

Look at the graphs below:

The figure has two graphs. In graph a there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (0, negative 1), (negative 1, 0), (1, 0), (negative 2, 3), and (2, 3). In graph b there is a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 2, 0), (2, 0), (0, negative 2), and (0, 2).

Graph a passes the vertical line test because if an imaginary vertical line were drawn anywhere on the graph, it would only touch one time.

Graph a is a function.

Graph b does not pass the vertical line test because when a vertical line is drawn down the graph, there is at least one place where the vertical line touches twice.

Graph b is not a function.

Try it

Try It: Determine if a Graph Is a Function

Determine if each graph is a function. Use the vertical line test to explain.

The figure has two graphs. In graph a there is an ellipse graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The ellipse goes through the points (0, negative 3), (negative 2, 0), (2, 0), and (0, 3). In graph b there is a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, negative 2), (2, 0), and (4, 2).

Compare your answser:

Here is how to determine if a graph is a function:

Graph a does not pass the vertical line test because if a vertical line were drawn down the graph, there would be at least one place where the vertical line would touch more than once. For example, a vertical line drawn at $x = 1$ would cross the graph more than once.

Graph a is not a function.

Graph b does pass the vertical line test because a vertical line drawn anywhere on the graph will only touch one time.

Graph b is a function.

4.5.2 Using the Vertical Line Test

Activity

Video: Interpreting Function Notation

Additional Resources

Determine If a Graph Is a Function

Try It: Determine if a Graph Is a Function