Continuity Graphs of basic functions.

1. 2.6 Graphs of Basic Functions
Chapter 2 Graphs and Functions

2. Concepts and Objectives
⚫ Graphs of Basic Functions
⚫ Continuity
⚫ Identifying the Identity, Squaring, Cubing, Square
Root, Cube Root, and Absolute Value function graphs
⚫ Graphing Piecewise functions, including the Greatest
Integer function

3. Continuity
⚫ Roughly speaking, a function is
continuous over an interval of its
domain if its hand-drawn graph
over the interval can be sketched
without lifing the pencil from the
paper.
⚫ If a function is not continuous at at
point, then it has a discontinuity
there.
Discontinuity
at (3, 1)

4. Graphs of Basic Functions
⚫ Graphs of the basic functions we study can be sketched
by plotting points or by using a program such as
Desmos.
⚫ Once you understand the shape of the basic graph, it is
much easier to understand what transformations do to
it.
⚫ You should already be familiar with most, if not all, of
these from either Algebra I or Algebra II.

5. Identity Function f(x) = x
⚫ f(x) = x is increasing on its entire domain, (–∞ ∞).
⚫ It is continuous on its entire domain, (–∞ ∞)
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
–2 –2
–1 –1
0 0
1 1
2 2

6. Squaring Function f(x) = x2
⚫ f(x) = x2 decreases on the interval (–∞ 0] and increases
on the interval [0, ∞).
⚫ It is continuous on its entire domain, (–∞ ∞)
Domain: (–∞ ∞) Range: [0 ∞)
x y
–2 4
–1 1
0 0
1 1
2 4
vertex

7. Cubing Function f(x) = x3
⚫ f(x) = x3 increases on its entire domain, (–∞ ∞).
⚫ It is continuous on its entire domain, (–∞ ∞)
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
–2 –8
–1 –1
0 0
1 1
2 8

8. Square Root Function
⚫ increases on its entire domain, [0 ∞).
⚫ It is continuous on its entire domain, [0 ∞)
Domain: [0 ∞) Range: [0 ∞)
x y
0 0
1 1
4 2
9 3
16 4
( )f x x=
( )f x x=

9. Cube Root Function
⚫ increases on its entire domain, (–∞ ∞).
⚫ It is continuous on its entire domain, (–∞ ∞)
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
-8 -2
-1 -1
0 0
1 1
8 2
( ) 3
f x x=
( ) 3
f x x=

10. Absolute Value Function
⚫ decreases on the interval (–∞ 0] and
increases on the interval [0, ∞).
⚫ It is continuous on its entire domain, (–∞ ∞)
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
-2 2
-1 1
0 0
1 1
2 2
( )f x x=
( )f x x=

11. Piecewise-Defined Functions
⚫ The absolute value function is defined by different rules
over different intervals of its domain. Such functions are
called piecewise-defined functions.
⚫ If you are graphing a piecewise function by hand, graph
each piece over its defined interval. If necessary, use
open and closed circles to mark discontinuities.
⚫ If you are using Desmos to graph a piecewise function,
you can control the interval graphed by putting braces
after the function.
⚫ You can make open circles by plotting the point and
changing the type of point used.

12. Piecewise-Defined Functions
⚫ Example: Graph the function.
( )
2 5 if 2
1 if 2
x x
f x
x x
− + 
= 
+ 

13. Greatest Integer Function
⚫ The greatest integer function, , pairs every
real number x with the greatest integer less than or
equal to x.
⚫ For example, 8.4 = 8, –5 = –5,  = 3, and –6.4 = –7.
⚫ In general, if , then
( )f x x=
( )f x x=
( )f x x=
( )
2 if 2 1
1 if 1 0
0 if 0 1 , etc.
1 if 1 2
2 if 2 3
x
x
f x x
x
x
− −   −
− −  

=  
  

 

14. Greatest Integer Function
⚫ is constant on the intervals …, [–2, –1), [–1, 0),
[0, 1), [1, 2), [2, 3), ….
⚫ It is discontinuous at all integers values in its entire
domain, (–∞ ∞).
Domain: (–∞ ∞) Range: {y | y ∊ ℤ}
x y
-2 -2
-0.5 -1
0 0
1 1
2.5 2
( )f x x=
( )f x x=

15. Greatest Integer Function
⚫ To graph this in Desmos, use the “floor” function. Make
one table of points with closed circles and one table with
open circles.
⚫ Example: Graph
( )f x x=
( )
1
1
2
f x x= +

16. The Relation x = y2
⚫ This is not a function, but you should see the relation
between it and the graphs of y = x2 and .
⚫ It is continuous on its entire domain, [0 ∞)
Domain: [0 ∞) Range: (–∞ ∞)
x y
0 0
1 –1
1 1
4 –2
4 2
y x=

17. Classwork
⚫ 2.6 Assignment (College Algebra)
⚫ Page 255: 2-20; page 242: 22-26, 36, 38;
page 227: 54-68
⚫ 2.6 Classwork Check
⚫ Quiz 2.5