Functions and Their Graphs Help (page 2)
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].
Increasing, Decreasing, and Constant Intervals
Increasing and Decreasing
A function is increasing on an interval if moving toward the right in the interval means the graph is going up. A function is decreasing on an interval if moving toward the right in the interval means the graph is going down. The function whose graph is in Figure 3.6 is increasing from x = −3 to x = 0 as well as to the right of x = 2. It is decreasing to the left of x = −3 and between x = 0 and x = 2. Using interval notation, we say the function is increasing on the intervals (−3, 0) and (2, ∞) and decreasing on the intervals (−∞, −3) and (0, 2). For reasons covered in calculus, parentheses are used for the interval notation.
A function is constant on an interval if the y -values do not change. This part of the graph will be part of a horizontal line.
Determine the intervals on which the functions are increasing, decreasing or constant.
This function is increasing on (−5, −2) and (4, 5). It is decreasing on (−2, 2) and constant on (2, 4).
The function is increasing on all of its domain, (0, ∞).
Find practice problems and solutions for these concepts at Functions and Their Graphs Practice Problems.
Find practice problems and solutions for these concepts at Functions and Their Graphs Practice Test.