Problem

1. Explain why S: x = y² is a NOT function.

2. Graph relation S and use the vertical line test to verify it is NOT a function.

3. State the domain and range of relation S.

Solution

1. Explain why S: x = y² is a NOT function.

Begin by solving for y in relation S: x = y².

The relation above is NOT a function because at least one x-value substituted into the relation results in more than one y-value. For a relation to be a function it must assign exactly one value to each element of a domain of values.

As an example, let x = 4.

Notice that two values result from letting x = 4. The two values are +2 and -2, or two ordered pairs ( 4, +2 ) and ( 4, -2 ).

Observe the animation illustrating the vertical line test below.

The vertical line test: If a vertical line can be drawn anywhere in the coordinate plane so that it intersects the graph of a relation at more than one location, then the relation is NOT a function.

In this example, a vertical line intersects the graph of the relation at more than one location for at least one value of x. As a result, the vertical line test indicates that the relation S (x = y²) is NOT a function.

3. State the domain and range of relation S.

Domain of S: { x | x ≥ 0 }

Range of S: { y | y ∈ R }