Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima.
Given an application, find a function that models the application. Find the domain of the function and function values.
Graph functions defined piecewise.
Because functions occur in so many real-world situations, it is important to be able to analyze them carefully.
On a given interval, if the graph of a function rises from left to right, it is said to be increasing on that interval. If the graph drops from left to right, it is said to be decreasing. If the function values stay the same on the interval, the function is said to be constant.
We are led to the following definitions.
Determine the intervals on which the function in the figure at left is
increasing;
decreasing;
constant.
When expressing interval(s) on which a function is increasing, decreasing, or constant, we consider only values in the domain of the function. Since the domain of this function is (−∞, ∞)
As x-values (that is, values in the domain) increase from x=3
As x-values increase from negative infinity to −1, y-values decrease; y-values also decrease as x-values increase from 5 to positive infinity. Thus the function is decreasing on the intervals (−∞, −1)
As x-values increase from −1 to 3, y remains −2. The function is constant on the interval (−1, 3).
Now Try Exercise 5.
In calculus, the slope of a line tangent to the graph of a function at a particular point is used to determine whether the function is increasing, decreasing, or neither. If the slope is positive, the function is increasing; if the slope is negative, the function is decreasing; if the slope is 0, the function is constant. Since slope cannot be both positive and negative at the same point, a function cannot be both increasing and decreasing at a specific point. For this reason, increasing, decreasing, and constant intervals are expressed in open-interval notation. In Example 1, if [3, 5] had been used for the increasing interval and [5, ∞)
Consider the graph shown below. Note the “peaks” and “valleys” at the x-values c1, c2
Simply stated, f(c)
If you take a calculus course, you will learn a method for determining exact values of relative maxima and minima. In Section 3.3, we will find exact maximum and minimum values of quadratic functions algebraically.
Using the graph shown below, determine any relative maxima or minima of the function f(x)=0.1x3−0.6x2−0.1x+2
We see that the relative maximum value of the function is 2.004. It occurs when x=−0.082
We note that the graph starts rising, or increasing, from the left and stops increasing at the relative maximum. From this point, the graph decreases to the relative minimum and then begins to rise again. The function is increasing on the intervals
and decreasing on the interval
Let’s summarize our results.
Relative Maximum | 2.004 at x=−0.082 |
Relative Minimum | −1.604 at x=4.082 |
Increasing | (−∞, −0.082), (4.082, ∞) |
Decreasing | (−0.082, 4.082) |
Now Try Exercise 15.
Many real-world situations can be modeled by functions.
Car Distance. Two nurses, Kiara and Matias, drive away from a hospital at right angles to each other. Kiara’s speed is 35 mph and Matias’s is 40 mph.
Express the distance between the cars as a function of time, d(t).
Find the domain of the function.
Suppose 1 hr goes by. At that time, Kiara has traveled 35 mi and Matias has traveled 40 mi. We can use the Pythagorean theorem to find the distance between them. This distance would be the length of the hypotenuse of a right triangle with legs measuring 35 mi and 40 mi. After 2 hr, the triangle’s legs would measure 2⋅35
After t hours, Kiara has traveled 35t miles and Matias 40t miles. We now use the Pythagorean theorem:
Because distance must be nonnegative, we need consider only the positive square root when solving for d(t):
Thus, d(t)=53.15t, t≥0
Since the time traveled, t, must be nonnegative, the domain is the set of nonnegative real numbers [0, ∞)
Now Try Exercise 25.
Area of Office Space. A community college has 30 ft of dividers with which to set off a rectangular area for a student testing center. If a corner of the math lab is used for the testing center, the partition need only form two sides of a rectangle.
Express the floor area of the office space as a function of the length of the partition.
Find the domain of the function.
Using the graph shown below, determine the dimensions that maximize the area of the floor.
Note that the dividers will form two sides of a rectangle. If, for example, 14 ft of dividers are used for the length of the rectangle, that would leave 30−14
The area, A(x)
The function A(x)=30x−x2
Because the rectangle’s length and width must be positive and only 30 ft of dividers are available, we restrict the domain of A to {x|0<x<30}
On the graph of the function shown at the top of the page, the maximum value of the area on the interval (0, 30) appears to be 225 when x=15
Length =x=15
Width =30−x=30−15=15
Now Try Exercise 31.
Sometimes functions are defined piecewise using different output formulas for different pieces, or parts, of the domain.
For the function defined as
find f(−5), f(−3), f(0), f(3), f(4)
First, we determine which part of the domain contains the given input. Then we use the corresponding formula to find the output.
Since −5<−2
Since −3<−2
Since −2≤0≤3
Since −2≤3≤3
Since 4>3
Since 10>3
Now Try Exercise 35.
Graph the function defined as
Since the function is defined in two pieces, or parts, we create the graph in two parts.
We graph g(x)=13x+3
x (x<3) |
g(x)=13x+3 |
---|---|
−3 | 2 |
0 | 3 |
2 | 323 |
We graph g(x)=−x
x (x≥3) |
g(x)=−x |
---|---|
3 | −3 |
4 | −4 |
6 | −6 |
Now Try Exercise 39.
Graph the function defined as
We create the graph in three pieces, or parts.
We graph f(x)=4
x (x≤0) |
f(x)=4 |
---|---|
−5 | 4 |
−2 | 4 |
0 | 4 |
We graph f(x)=4−x2
x (0<x≤2) |
f(x)=4−x2 |
---|---|
12 |
334 |
1 | 3 |
2 | 0 |
We graph f(x)=2x−6
x (x>2) |
f(x)=2x−6 |
---|---|
212 |
−1 |
3 | 0 |
5 | 4 |
Now Try Exercise 43.
Graph the function defined as
When x≠−2
Thus,
The graph of this part of the function consists of a line with a “hole” at the point (−2, −4)
Now Try Exercise 47.
A piecewise function with importance in calculus and computer programming is the greatest integer function, f, denoted f(x)=〚x〛
The greatest integer function pairs each input with the greatest integer less than or equal to that input. Thus x-values 1, 112
Graph f(x)=〚x〛
The greatest integer function can also be defined as a piecewise function with an infinite number of statements.
We see that the domain of this function is the set of all real numbers, (−∞, ∞)
Now Try Exercise 51.
Determine the intervals on which the function is
increasing;
decreasing;
constant.
1.
2.
3.
4.
5.
6.
7.–12. Determine the domain and the range of each of the functions graphed in Exercises 1–6.
Using the graph, determine any relative maxima or minima of the function and the intervals on which the function is increasing or decreasing.
13.
14.
15.
16.
Graph the function. Estimate the intervals on which the function is increasing or decreasing and any relative maxima or minima.
17. f(x)=x2
18. f(x)=4−x2
19. f(x)=5−|x|
20. f(x)=|x+3|−5
21. f(x)=x2−6x+10
22. f(x)=−x2−8x−9
23. Lumberyard. Rick’s lumberyard has 480 yd of fencing with which to enclose a rectangular area. If the enclosed area is x yards long, express its area as a function of its length.
24. Triangular Flag. A seamstress is designing a triangular flag so that the length of the base of the triangle, in inches, is 7 less than twice the height h. Express the area of the flag as a function of the height.
25. Blimp Distance. The Goodyear Blimp can be seen flying at an altitude of 3500 ft above the Motor Speedway during the Indianapolis 500 race. The slanted distance directly to the Pagoda at the start–finish line is d feet. Express the horizontal distance h as a function of d.
26. Rising Balloon. A hot-air balloon rises straight up from the ground at a rate of 120 ft/min. The balloon is tracked from a rangefinder on the ground at point P, which is 400 ft from the release point Q of the basket. Let d=
27. Inscribed Rhombus. A rhombus is inscribed in a rectangle that is w meters wide with a perimeter of 40 m. Each vertex of the rhombus is a midpoint of a side of the rectangle. Express the area of the rhombus as a function of the width of the rectangle.
28. Carpet Area. A carpet installer uses 46 ft of linen tape to bind the edges of a rectangular hall runner. If the runner is w feet wide, express its area as a function of the width.
29. Golf Distance Finder. A device used in golf to estimate the distance d, in yards, to a hole measures the size s, in inches, that the 7-ft pin appears to be in a viewfinder. Express the distance d as a function of s.
30. Gas Tank Volume. A gas tank has ends that are hemispheres of radius r feet. The cylindrical midsection is 6 ft long. Express the volume of the tank as a function of r.
31. Swimming Areas. A summer camp has 240 ft of float line with which to rope off three adjacent rectangular areas of a lake for swimming lessons, one for each of three levels of swimming ability. A beach forms one side of the swimming areas. Suppose the width of each area is x yards.
Express the total area of the three swimming areas as a function of x.
Find the domain of the function.
Using the graph of the function shown below, determine the dimensions that yield the maximum area.
32. Play Space. A car dealership has 24 ft of dividers with which to enclose a rectangular play space in a corner of a customer lounge. The sides against the wall require no partition. Suppose the play space is x feet long.
Express the area of the play space as a function of x.
Find the domain of the function.
Using the graph shown below, determine the dimensions that yield the maximum area.
33. Volume of a Box. From a 12-cm by 12-cm piece of cardboard, square corners are cut out so that the sides can be folded up to make a box.
Express the volume of the box as a function of the side x, in centimeters, of a cut-out square.
Find the domain of the function.
Using the graph of the function shown below, determine the dimensions that yield the maximum volume.
34. Office File. Designs Unlimited plans to produce a one-component vertical file by bending the long side of an 8-in. by 14-in. sheet of plastic along two lines to form a shape.
Express the volume of the file as a function of the height x, in inches, of the file.
Find the domain of the function.
Using the graph of the function shown below, determine how tall the file should be in order to maximize the volume that the file can hold.
For each piecewise function, find the specified function values.
35. g(x)={x+4,for x≤1,8−x,for x>1
g(−4), g(0), g(1)
36. f(x)={3,for x≤−2,12x+6,for x>−2
f(−5), f(−2), f(0)
37. h(x)={−3x−18,for x<−5,1,for −5≤x<1,x+2,for x≥1
h(−5), h(0), h(1)
38. f(x)={−5x−8,for x<−2,12x+5,for −2≤x≤4,10−2x,for x>4
f(−4), f(−2), f(4)
Graph each of the following.
39. f(x)={12x,for x<0,x+3,for x≥0
40. f(x)={−13x+2,for x≤0,x−5,for x>0
41. f(x)={−34x+2,for x<4,−1,for x≥4
42. h(x)={2x−1,for x<2,2−x,for x≥2
43. f(x)={x+1,for x≤−3,−1,for −3<x≥4,12x,for x≥4
44. f(x)={4,for x≤−2,x+1,for −2<x<3,−x,for x≥3
45. g(x)={12x−1,for x<0,3,for 0≤x<1,−2x,for x>1
46. f(x)={x2−9x+3,for x≠−3,5,for x=−32,for x=5,
47. f(x)={2,for x=5,x2−25x−5,for x≠5
48. f(x)={x2+3x+2x+1,for x≠−1,7,for x=−1
49. f(x)=〚x〛
50. f(x)=2〚x〛
51. g(x)=1+〚x〛
52. h(x)=12〚x〛−2
53.–58. Find the domain and the range of each of the functions defined in Exercises 39–44.
Determine the domain and the range of the piecewise function. Then write an equation for the function.
59.
60.
61.
62.
63.
64.
65. Given f(x)=5x2−7
f(−3)
f(3)
f(a)
f(−a)
66. Given f(x)=4x3−5x
f(2)
f(−2)
f(a)
f(−a)
67. Write an equation of the line perpendicular to the graph of the line 8x−y=10
68. Find the slope and the y-intercept of the line with equation 2x−9y+1=0
69. Parking Costs. A parking garage charges $3 for up to (but not including) 1 hr of parking, $6 for up to 2 hr of parking, $9 for up to 3 hr of parking, and so on. Let C(t)=
Graph the function.
Write an equation for C(t)
70. If 〚x+2〛=−3
71. If (〚x〛)2=25
72. Minimizing Power Line Costs. A power line is constructed from a power station at point A to an island at point I, which is 1 mi directly out in the water from a point B on the shore. Point B is 4 mi downshore from the power station at A. It costs $5000 per mile to lay the power line under water and $3000 per mile to lay the power line under ground. The line comes to the shore at point S downshore from A. Let x=
Express the cost C of laying the line as a function of x.
At what distance x from point B should the line come to shore in order to minimize cost?
73. Volume of an Inscribed Cylinder. A right circular cylinder of height h and radius r is inscribed in a right circular cone with a height of 10 ft and a base with radius 6 ft.
Express the height h of the cylinder as a function of r.
Express the volume V of the cylinder as a function of r.
Express the volume V of the cylinder as a function of h.