Answer: The most common graphs name the input value
x
and the output value
y
, and we say
y
is a function of
x
, or
y
=
f
(
x
)
when the function is named
f
. The graph of the function is the set of all points
(
x
,
y
)
in the plane that satisfies the equation
y
=
f
(
x
)
. If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in the graph below tell us that
f
(
0
)
=
2
and
f
(
6
)
=
1
. However, the set of all points
(
x
,
y
)
satisfying
y
=
f
(
x
)
is a curve. The curve shown includes
(
0
,
2
)
and
(
6
,
1
)
because the curve passes through those points.
Graph of a polynomial.
The vertical line test can be used to determine whether a graph represents a function. A vertical line includes all points with a particular
x
value. The
y
value of a point where a vertical line intersects a graph represents an output for that input
x
value. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because that
x
value has more than one output. A function has only one output value for each input value.
Three graphs visually showing what is and is not a function.
HOW TO: GIVEN A GRAPH, USE THE VERTICAL LINE TEST TO DETERMINE IF THE GRAPH REPRESENTS A FUNCTION.
Inspect the graph to see if any vertical line drawn would intersect the curve more than once.
If there is any such line, the graph does not represent a function.
If no vertical line can intersect the curve more than once, the graph does represent a function.
Explanation: