Pat05 ppt 0201

1. Copyright © 2016, 2012 Pearson Education, Inc. 2 -1
Chapter 2
More on
Functions

2. Copyright © 2016, 2012 Pearson Education, Inc. 2 -2
Section
2.1 Increasing, Decreasing, and Piecewise
Functions; Applications
2.2 The Algebra of Functions
2.3 The Composition of Functions
2.4 Symmetry
2.5 Transformations
2.6 Variation and Applications

3. Copyright © 2016, 2012 Pearson Education, Inc. 2 -3
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
minima.
 Given an application, find a function that models the
application. Find the domain of the function and
function values.
 Graph functions defined piecewise.

4. Copyright © 2016, 2012 Pearson Education, Inc. 2 -4
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.

5. Copyright © 2016, 2012 Pearson Education, Inc. 2 -5
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).

6. Copyright © 2016, 2012 Pearson Education, Inc. 2 -6
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).

7. Copyright © 2016, 2012 Pearson Education, Inc. 2 -7
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).
Definitions continued

8. Copyright © 2016, 2012 Pearson Education, Inc. 2 -8
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.

9. Copyright © 2016, 2012 Pearson Education, Inc. 2 -9
Relative Maximum and Minimum Values
y
xc1 c2 c3
Relative
minimum
Relative
maximum f

10. Copyright © 2016, 2012 Pearson Education, Inc. 2 -10
Applications of Functions
Many real-world 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.
Now A(w) = (40 – w)w = 40w – w2.

11. Copyright © 2016, 2012 Pearson Education, Inc. 2 -11
Functions Defined Piecewise
For the function defined as:
find f (-3), f (1), and f (5).
2
, for 0,
( ) 4, for 0 2,
1, for 2,
x x
f x x
x x
 

  
  
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.



12. Copyright © 2016, 2012 Pearson Education, Inc. 2 -12
c) We graph f(x) = only for
inputs x greater than 2.
Functions Defined Piecewise
Graph the function defined as:
a) We graph f(x) = 3 only for
inputs x less than or equal to 0.
b) We graph f(x) = 3 + x2 only
for inputs x greater than 0 and
less than or equal to 2.
2
3 for 0
( ) 3 for 0 2
1 for 2
2
x
f x x x
x
x

 

    

  

f(x) = 3, for x  0
f(x) = 3 + x2, for 0 < x  2
( ) 1for 2
2
x
f x x  
1
2
x


13. Copyright © 2016, 2012 Pearson Education, Inc. 2 -13
Functions Defined Piecewise
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.
2
4
for 2
( ) 2
3, for 2
 
 
 
  
x
x
f x x
x
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.

14. Copyright © 2016, 2012 Pearson Education, Inc. 2 -14
= the greatest integer less than or
equal to x.
Greatest Integer Function
5
5.2
1
5
8



The greatest integer function pairs the input with
the greatest integer less than or equal to that
input.
–5
0
0.2
1
7
0
3
3.2
1
3
8
3
( ) f x x