171S2.1 Increasing, Decreasing, and Piecewise Functions; Applications September 16, 2010
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 2: More on Functions
2.1 Increasing, Decreasing, and Piecewise Functions; Applications
2.2 The Algebra of Functions
2.3 The Composition of Functions
2.4 Symmetry and Transformations 2.5 Variation and Applications
Increasing, Decreasing, and Constant Functions
On a given open interval, If the graph of a function rises from left to right, it is said to be increasing on that interval. 2.1 Increasing, Decreasing, and Piecewise Functions; Applications
• Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and relative minima.
• Given an application, find a function that models the application; find the domain of the function and function values, and then graph the function.
• Graph functions defined piecewise.
See the piecewise function animation in Course Documents of CourseCompass.
Relative Maximum and Minimum Values
Suppose that f is a function for which f(c) exists for some c in the domain of f. Then:
y
If the graph drops from left to right, it is said to be decreasing. If the function values stay the same from left to right, the function is said to be constant.
Relative maximum
f(c) is a relative maximum if there exists an open interval I containing c such that f(c) > f(x), for all x in I where x ≠ c; and f(c) is a relative minimum if there exists an open interval I containing c such that f(c) < f(x), for all x in I where x ≠ c.
Relative minimum
c1
c2
c3
x
1
171S2.1 Increasing, Decreasing, and Piecewise Functions; Applications September 16, 2010
Applications of Functions Many real­world situations can be modeled by functions.
Example
A man plans to enclose a rectangular area using 80 yards of fencing. If the area is w yards wide, express the enclosed area as a function of w.
Solution
We want area as a function of w. Since the area is rectangular, we have A = lw. We know that the perimeter, 2 lengths and 2 widths, is 80 yds, so we have 40 yds for one length and one width. If the width is w, then the length, l, can be given by l = 40 – w. Functions Defined Piecewise
Some functions are defined piecewise using different output formulas for different parts of the domain.
For the function defined as:
find f (­3), f (1), and f (5).
Since –3 < 0, use f (x) = x2: f (–3) = (–3)2 = 9. Since 0 < 1 < 2, use f (x) = 4: f (1) = 4.
Now A(w) = (40 – w)w = 40w – w2.
Since 5 > 2 use f (x) = x – 1: f (5) = 5 – 1 = 4. Functions Defined Piecewise
Greatest Integer Function
Graph the function defined as:
= the greatest integer less than or equal to x.
The greatest integer function pairs the input with the greatest integer less than or equal to that input.
a) We graph f(x) = ­3 only for inputs x < 0.
b) We graph f(x) = ­3 + x2 only for inputs 0 < x < 2.
c) We graph f(x) = only for inputs x > 2.
2
171S2.1 Increasing, Decreasing, and Piecewise Functions; Applications September 16, 2010
175/6. Determine the intervals on which the function is (a) increasing, (b) decreasing, and (c) constant.
176/25. Graph the function using the given viewing window. Find the intervals on which the function is increasing or decreasing and find any relative maxima or minima. Change the viewing window if it seems appropriate for further analysis.
f(x) = 1.1x4 ­ 5.3 x2 + 4.07 [­4, 4, ­4, 8]
175/16. Using the graph, determine any relative maxima or minima of the function and the intervals on which the function is increasing or decreasing.
179/46. Cost of Material. A rectangular box with volume 320 ft3 is built with a square base and top. The cost is $1.50 / ft2 for the bottom, $2.50 / ft2 for the sides, and $1 / ft2 for the top. Let x = the length of the base, in feet.
a) Express the cost of the box as a function of x. b) Find the domain of the function. c) Graph the function with a graphing calculator. d) What dimensions minimize the cost of the box?
3
171S2.1 Increasing, Decreasing, and Piecewise Functions; Applications September 16, 2010
This graph with Zoomfit represents the cost function. Considering the domain 0 < x < √320, we are interested only in a portion of the first quadratic of the graph. Set the window as indicated below and find the relative minimum point as shown .
(b)
179/58. Make a hand­drawn graph of each of the following. Check your results using a graphing calculator.
When x = 8.6177404, y = cost = 556.99066.
180/74. Determine the domain and the range of the piecewise function. Then write an equation for the function.
4