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# 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 x2 - y2 = 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 x2 - y2 = 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 x2 - y2 = 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 x2 - y2 = 1. If x2 - y2 = 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, x2 - y2 = 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 - 2y4x - 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) = x2 + 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, x2 + 5). It is also the solution set of y = x2 + 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 = x2, we can solve for y by subtracting x from each side to get y = x2 - x.

This equation now gives a function. If we name the function f, then the function is defined by f(x) = x2 - x.