1. Copyright © 2015, 2011, 2005 Pearson Education, Inc.
Graphs and Functions
Copyright © 2015, 2011, 2005 Pearson Education, Inc. 1

2. 2.4 A Library of Parent Functions Continuity
The Identity, Squaring, and Cubing Functions
The Square Root and Cube Root Functions
The Absolute Function
Piecewise-Defined Functions
The Relation x = y2

3. Continuity (Informal Definition)
A function is continuous over an interval of its domain if its hand-drawn graph over that interval can be sketched without lifting the pencil from the paper.

4. Describe the intervals of continuity for each function.
DETERMINING INTERVALS OF CONTINUTIY
Example 1
Describe the intervals of continuity for each function.
Solution The function is continuous over its entire domain,(– , ).

5. Describe the intervals of continuity for each function.
DETERMINING INTERVALS OF CONTINUTIY
Example 1
Describe the intervals of continuity for each function.
Solution The function has a point of discontinuity at
x = 3. Thus, it is continuous over the intervals ,
(– , 3) and (3, ). 3

6. IDENTITY FUNCTION  (x) = x x y – 2 – 1 1 2
Domain: (– , ) Range: (– , )
IDENTITY FUNCTION  (x) = x y x y
– 2
– 1 1 2 x
(x) = x is increasing on its entire domain, (– , ).
It is continuous on its entire domain.

7. SQUARING FUNCTION  (x) = x2 x y – 2 4 – 1 1
Domain: (– , ) Range: [0, )
SQUARING FUNCTION  (x) = x2 y x y
– 2 4
– 1 1 2 x
(x) = x2 decreases on the interval (– ,0] and increases on the interval [0, ).
It is continuous on its entire domain, (– , ).

8. CUBING FUNCTION  (x) = x3 x y – 2 – 8 – 1 1 2 8
Domain: (– , ) Range: (– , )
CUBING FUNCTION  (x) = x3 y x y
– 2
– 8
– 1 1 2 8 x
(x) = x3 increases on its entire domain, (– ,).
It is continuous on its entire domain, (– , ).

9. SQUARE ROOT FUNCTION  (x) = x y 1 4 2 9 3 16
Domain: [0, ) Range: [0, )
SQUARE ROOT FUNCTION  (x) = y x y 1 4 2 9 3 16 x
(x) = increases on its entire domain, [0,).
It is continuous on its entire domain, [0, ).

10. CUBE ROOT FUNCTION  (x) = x y – 8 – 2 – 1 1 8 2
Domain: (– , ) Range: (– , )
CUBE ROOT FUNCTION  (x) = y x y
– 8
– 2
– 1 1 8 2 x
(x) = increases on its entire domain, (– , ).
It is continuous on its entire domain, (– , ).

11. ABSOLUTE VALUE FUNCTION  (x) = x y – 2 2 – 1 1
Domain: (– , ) Range: [0, )
ABSOLUTE VALUE FUNCTION  (x) = y x y
– 2 2
– 1 1 x
(x) = decreases on the interval (– , 0] and increases on [0, ).
It is continuous on its entire domain, (– , ).

12. Graph each function. (a) (b) GRAPHING PIECEWISE-DEFINED FUNCTIONS
Example 2
Graph each function.
(a)
(b)

13. Solution Graph each interval of the domain separately.
GRAPHING PIECEWISE-DEFINED FUNCTIONS
Example 2
Graph each function. y
(a) 2 4 6
– 2 3 5
(2, 3)
(2, 1)
Solution Graph each interval of the domain separately. x

14. The two graphs meet at the point (0,3).
GRAPHING PIECEWISE-DEFINED FUNCTIONS
Example 2
Graph the function.
(b)
Solution
The two graphs meet at the point (0,3).

16. Copyright © 2015, 2011, 2005 Pearson Education, Inc.
Domain: (– , )
Range: {y  y is an integer} = {…–3, – 2, – 1, 0, 1, 2, 3,…}
GREATEST INTEGER FUNCTION  (x) = x y
– 2
– 1.5
–0.99
– 1
0.001 3
3.99 1 2 3
– 2 4
– 3
– 4
(x) = is constant on the intervals…, [– 2, – 1), [– 1, 0), [0, 1), [1, 2), [2, 3),….
It is discontinuous at all integer values in its domain (– , ). 16
Copyright © 2015, 2011, 2005 Pearson Education, Inc.

17. GRAPHING A GREATEST INTEGER FUNCTION
Example 3
Graph
Solution If x is in the interval [0, 2), then y = 1. For x in [2, 4), y = 2, and so on. Some sample ordered pairs are given here. x
1/2 1
3/2 2 3 4
– 1
– 2
– 3 y –1
The ordered pairs in the table suggest the graph shown. The domain is (– , ). The range is {…, – 2, – 1, 0, 1, 2,…}.

18. APPLYING A GREATEST INTEGER FUNCTION
Example 4
An express mail company charges $25 for a package weighing up to 2 lb. For each additional pound or fraction of a pound there is an additional charge of $3. Let y = D(x) represent the cost to send a package weighing x pounds. Graph y = D(x) for x in the interval (0, 6].

19. APPLYING A GREATEST INTEGER FUNCTION Example 420
Pounds 1 2 3 4 5 y 30 40 6 x
Solution
For x in the interval
(0, 2], y = 25.
For x in (2, 3],
y = = 28.
For x in (3, 4],
y = = 31, and so on.
Dollars

20. The Relation x = y2
Recall that a function is a relation where every domain is paired with one and only one range value. x y 1 1 4 2 9 3 y x
x = y2
Note that this is a relation, but not a function.
Domain is [0, ). Range is (– , ).