Equations as Functions
Equations that Define Relations
Suppose we have an equation in two variables x and y. Letâ€™s
consider y = 4x + 3 and
x^{2}  y^{2} = 1.
For any equation, the solution set is the set of ordered pairs (x, y)
for which the equation holds true when the values are substituted
for x and y. Since a set of ordered pairs is a relation, we can
conclude that a solution set is a relation. In this way, equations
represent relations.
A relation that can be represented by an equation can easily be
visualized by graphing the equation. The graphs of y = 4x + 3 and
x^{2}  y^{2} = 1 are shown.
When is a Relation a Function?
The set of first coordinates in a relation is called the domain of the
relation. The set of second coordinates is called the range.
Remember that a relation is a function if each element of the
domain is paired with exactly one element in the range.
When we have a relation defined by an equation, we can graph the
equation and then visually determine whether or not it is a function
by using the vertical line test.
Key Idea
For a relation defined by an equation, the relation is a function if
every vertical line intersects the graph of the equation in at most
one point.
Consider the graph of y = 4x + 3. Using the vertical line test, we find
that the equation does define a function.
As the vertical line moves to
the right across the graph, it
intersects only one point at a
time. 

Consider the graph of
x^{2}  y^{2} = 1. Using the vertical line test, we
find that the equation does not define a function.
As the vertical line moves to
the right across the graph, it
usually intersects the graph in
two points. Sometimes it does
not intersect the graph at all. 

Recognizing Functions Algebraically
Another way to recognize when an equation defines a function is to
solve for y in terms of x.
Key Idea Any equation which has y on one side and a single formula
involving x on the other side defines a function.
The following equations are examples of functions.
Some equations are not functions. Consider x^{2}  y^{2} = 1. If x^{2}
 y^{2} = 1 is solved for y, the result is
The symbol
Â± indicates
that there are two formulas on the right side of the equation for y.
Hence, x^{2}  y^{2} = 1 does not represent a function.
If an equation is not written in the form y = (formula in x), the
equation may still determine a function. Consider the following
example.
In 4x  2y = 10, y is not isolated on the left side. However, we can
solve the equation for y.
4x  2y 4x  2y + 2x
2y 
= 10 = 10  4x
=  4x + 10 
Subtract 4x from each side. 


Divide each side by 2. 
y 
= 2x  5 

This is now in the form y = (formula in x) and gives a function.
Key Idea If, in an equation, we can solve for y as a single formula in terms
of x, then that equation represents a function.
Functions as Rules
Functions can also be thought of as rules that take an input value of x and
produce an output value of y. We often give these rules names such as f, g, h,
etc. We can then define a function by a formula in x.
The rule f(x) = x^{2} + 5 represents a function. We can think of it as a relation by taking the function
to be the set of all ordered pairs of the form (x, f(x)) = (x, x^{2} + 5). It is
also the solution set of y = x^{2} + 5.
Often we are given an equation in which y is not given as a formula
in x, but we can solve for y in terms of x to get a rule defining the
function.
In y + x = x^{2}, we can solve for y by subtracting x from each side
to get
y = x^{2}  x.
This equation now gives a function. If we name the function f, then
the function is defined by
f(x) = x^{2}  x.
