Functions and Their Graphs Help
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.
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 y2 = 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 y2 = 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.
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].