This document provides information about graphing basic functions such as the identity, squaring, cubing, square root, cube root, and absolute value functions. It discusses concepts of continuity and discontinuity. It also describes how to graph piecewise functions, including the greatest integer function, using Desmos. Key functions and their domains, ranges, and graphs are defined.

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