1. Increasing, Decreasing, and Piecewise Functions; Applications
Section 2.1
Increasing, Decreasing, and Piecewise Functions; Applications

2. Objectives
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, and then graph the function.
Graph functions defined piecewise.

3. Increasing, Decreasing, and Constant Functions
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 from left to right, the function is said to be constant.

4. Definitions
A function f is said to be increasing on an open interval I, if for all a and b in that interval, a < b implies f(a) < f(b).

5. Definitions continued
A function f is said to be decreasing on an open interval I, if for all a and b in that interval, a < b implies f(a) > f(b).

6. Definitions continued
A function f is said to be constant on an open interval I, if for all a and b in that interval, f(a) = f(b).

7. 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: 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.

8. Relative Maximum and Minimum Values

9. Example
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.

10. Example continued
Solution Suppose that 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. The 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 70 mi, and 80 mi. Noting that the distances will always be changing, we make a drawing and let t = time, in hours, that Kiara and Matias have been driving since leaving the hospital.

11. Example continued
After t hours, Kiara has traveled 35t miles and Matias 40t miles. We now use the Pythagorean theorem:

12. Example continued
Because distance must be nonnegative, we need to consider only the positive square root when solving for d(t): Thus, d(t) = 53.15t, t ≥ 0. b.) Since the time traveled, t must be nonnegative, the domain is the set of nonnegative real numbers [0, ∞).

13. Example
Some functions are defined piecewise using different output formulas for different parts of the domain. For the function defined as: find f (−5), f (−3), f(0), f(3), f(4), and f(10)
Since –5 < − 2, use f (x) = f(x) = x + 1: f(−5) = − = − 4.
Since –3 < 2, use f (x) = x + 1: f (− 3) = − = − 2.
Since − 2 ≤ 0 ≤ 3, use f (x) = 5: f (0) = 5.

14. Example continued Since − 2 ≤ 3 ≤ 3, use f(x) = 5: f (3) = 5.
Since 4 > 3, use f(x) = x2: f(4) = 42 = 16.
Since 10 > 3, we once again use f(x) = x2: f(10) = 102 = 100.

15. Example We graph g(x) = 1/3x + 3 only for inputs less than 3.
Graph the function defined as:
We graph g(x) = 1/3x + 3 only for inputs less than 3.
We graph g(x) = − x only for inputs x greater than or equal to 3.

16. Example Graph the function defined as: Thus, f(x) = x – 2, for x ≠ ‒2
The graph of this part of the function consists of a line with a “hole” at (‒2, ‒4), indicated by the open circle.
The hole occurs because a piece of the function is not defined for x = ‒2.
f(‒2) = 3, so plot the point (‒2, 3) above the open circle.

17. Greatest Integer Function
= 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. –4 −1