1
c Roberto Barrera, Fall 2015
Math 142 1.1 Increasing, Decreasing, Concavity, Continuity, and Piecewise-Defined Functions
Increasing, Decreasing, Concavity, and Continuity
Increasing:
Decreasing:
Concave Up:
Concave Down:
Continuous:
Example: Determine where (use interval notation) the following function is increasing, decreasing, concave up,
concave down, and continuous.
3
2.5
2

2x,



3,
f (x) =
−x + 3,



−(x − 3)3 + 2,
1.5
1
0.5
0.5
1
1.5
2
2.5
3
0≤x<1
x=1
1<x≤2
2<x≤3
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c Roberto Barrera, Fall 2015
Math 142 Piecewise-Defined Function - A function that is defined by different rules for different parts of its domain.
Example: Let
f (x) =

 (x + 1)2 −4 ≤ x < 1

−x + 9
x≥1
a) Find f (−2) and f (1).
b) Graph f (x).
Example: Find the domain of the following function:
 x−5


 x+2
f (x) =


 x
x−4
x ≤ −3
x > −3
Example: Write f (x) = |x − 8| as a piecewise-defined function.
HINT: Set the quantity in absolute value signs ≥ 0 (or > 0 if the quantity is in the denominator), and solve for x
(this will tell you where the quantity is nonnegative).
3
c Roberto Barrera, Fall 2015
Math 142 Example: Write g(x) =
x−5
as a piecewise-defined function.
3|2 − x|
Example: A cell phone company has a base charge of $20/month. The first 100 minutes are free, and the next 400
minutes cost $0.10/minute. Any usage over 500 minutes costs $0.15/minute. Find a function, C(x), which gives
the amount of a cell phone bill during a month when a customer uses x cell phone minutes.
√
(x + 3) 4 x − 5
Review Example: Find the domain of f (x) = 2
.
x − x − 12
HINT (domain rules thus far):
1. Denominator 6= 0.
2. Cannot take the even root (in this case, the 4th root) of a negative number.