Introduction to Functions and Their Graphs
The graph of a function can give us a great deal of information about the function. In this chapter we will use the graph of a function to evaluate the function, find the x and y intercepts (if any), the domain and range, and determine where the function is increasing or decreasing (an important idea in calculus).
To say that f (−3) = 1 means that the point (−3, 1) is on the graph of f (x). If (5, 4) is a point on the graph of f (x), then f (5) = 4.
Example

The graph in Figure 3.1 is the graph of f ( x ) = x ^{3} − x ^{2} − 4 x + 4. Find f (−1), f (0), f (3), and f (−2).
The point (−1, 6) is on the graph means that f (−1) = 6.
The point (0, 4) is on the graph means that f (0) = 4.
The point (3, 10) is on the graph means that f (3) = 10.
The point (−2, 0) is on the graph means that f (−2) = 0.
Intercepts of a Graph
The graph also shows the intercepts of the graph. Remember that an x intercept is a point where the graph touches the x axis, and the y intercept is a point where the graph touches the y axis. We can tell that the y intercept for the graph in Figure 3.1 is 4 (or (0, 4)) and the x intercepts are −2, 1, and 2 (or (−2, 0), (1, 0) and (2, 0)).
The Vertical Line Test
An equation “gives y as a function of x ” means that for every x value, there is a unique y value. From this fact we can look at a graph of an equation to decide if the equation gives y as a function of x . If an x value has more than one y value in the equation, then there will be more than one point on the graph that has the same x coordinate. A line through points that have the same x coordinate is vertical. This is the idea behind the Vertical Line Test . The graph of an equation passes the Vertical Line Test if every vertical line touches the graph at one point or not at all. If so, then the equation is a function.
The graph of y^{2} = x is shown in Figure 3.2. The vertical line x = 4 touches the graph in two places, (4, 2) and (4, −2), so y is not a function of x in the equation y^{2} = x .
The Domain and Range of a Function
The domain of a function consists of all possible x values. We can find the domain of a function by looking at its graph. The graph’s extension horizontally shows the function’s domain. The range of a function consists of all possible y values. The graph’s vertical extention shows the function’s range.
Examples
Give the domain and range in interval notation.

The graph extends horizontally from x = −5 to x = 4. Because there are closed dots on these endpoints (instead of open dots), x = − 5 and x = 4 are part of the domain, too. The domain is [−5, 4]. The graph extends vertically from y = −4 to y = 3. The range is [−4, 3].

The graph extends horizontally from x = −3 to x = 2. Because open dots are used on (−3, 5) and (2, 0), these points are not on the graph, so x = −3 and x = 2 are not part of the domain. The domain is (−3, 2). The graph extends vertically from y = −4 and y = 5. The range is [−4, 5). We need to use a bracket around −4 because (0, −4) is a point on the graph, and a parenthesis around 5 because the point (−3, 5) is not a point on the graph.

The graph extends horizontally from x = − 2 on the left and vertically from below y = 0. The domain is [−2, ∞), and the range is (−∞, 0].

1
 2
Ask a Question
Have questions about this article or topic? AskPopular Articles
 Kindergarten Sight Words List
 First Grade Sight Words List
 10 Fun Activities for Children with Autism
 Grammar Lesson: Complete and Simple Predicates
 Definitions of Social Studies
 Child Development Theories
 Signs Your Child Might Have Asperger's Syndrome
 Social Cognitive Theory
 How to Practice Preschool Letter and Name Writing
 Theories of Learning