2.1 Increasing, Decreasing, and - Cape Fear...

171S2.1 Increasing, Decreasing, and Piecewise Functions; Applications

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CHAPTER 2: More on Functions2.1 Increasing, Decreasing, and Piecewise Functions; Applications2.2 The Algebra of Functions2.3 The Composition of Functions2.4 Symmetry and Transformations 2.5 Variation and Applications

MAT 171 Precalculus AlgebraDr. Claude Moore

Cape Fear Community College

Explanations and Exploratory Exercises for Interactive Figures http://media.pearsoncmg.com/aw/aw_bittinger_colalgebra_5/hidden/bca05_explorations.html

Interactive Figures from Pearson Education: Increasing and Decreasing Functions Graphs of Piecewise Functions

http://media.pearsoncmg.com/aw/aw_bittinger_colalgebra_5/bca05_ifig_launch.html

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.




Increasing, Decreasing, and Constant FunctionsOn a given open 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 from left to right, the function is said to be constant.




Relative Maximum and Minimum ValuesSuppose that f is a function

for which f(c) exists for some c in the domain of f. Then:

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.

y

c1 c2 c3

Relative minimum

Relative maximum

x

http://media.pearsoncmg.com/aw/aw_bittinger_colalgebra_5/hidden/bca05_explorations.html







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Applications of Functions Many realworld situations can be modeled by functions.

ExampleA 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.

Now A(w) = (40 – w)w = 40w – w2.

Functions Defined Piecewise

For the function defined as:find f (3), f (1), and f (5).

Some functions are defined piecewise using different output formulas for different parts of the domain.

Since –3 < 0, use f (x) = x2: f (–3) = (–3)2 = 9.

Since 0 < 1 < 2, use f (x) = 4: f (1) = 4.

Since 5 > 2 use f (x) = x – 1: f (5) = 5 – 1 = 4.

c) We graph f(x) = only for inputs x > 2.

Functions Defined PiecewiseGraph the function defined as:

a) We graph f(x) = 3 only for inputs x < 0.

b) We graph f(x) = 3 + x2 only for inputs 0 < x < 2.




= the greatest integer less than or equal to x.

Greatest Integer Function

The greatest integer function pairs the input with the greatest integer less than or equal to that input.




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175/6. Determine the intervals on which the function is (a) increasing, (b) decreasing, and (c) constant.

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.

f(x) = 0.09x3 + 0.5x2 0.1x + 1

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]

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?

The minimum cost would be $557.00 when x = 8.6 ft.


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179/58. Make a handdrawn graph of each of the following. Check your results using a graphing calculator.

Closed dot at (3, 5)

Open circleat (3, 6)

The fraction reduces to f(x) = x + 3 when x ≠ 3.

180/74. Determine the domain and the range of the piecewise function. Then write an equation for the function.

Domain: (∞, ∞)Range: [2, 2] ∪ [0, ∞)

Equation: It could be any of following

2.1 Increasing, Decreasing, and - Cape Fear...

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