Continuity
Informal Definition: Continuity –
Def.: A function 𝑦 = 𝑓(𝑥) is continuous at 𝑎 if:
(i) 𝑓(𝑎) exists
(ii) as 𝑥 → 𝑎 from the left, 𝑓(𝑥) → 𝑏 and
as 𝑥 → 𝑎 from the right, 𝑓(𝑥) → 𝑏.
(iii) 𝑏 = 𝑓(𝑎)
Def.: A function is continuous on an open interval if it is continuous at all points on
that interval.
Notes: 1. Asymptotes in a graph indicate discontinuity.
2. ALL polynomial functions are continuous.
Ex.: Determine if each graph is continuous.
1
Discontinuity of Functions
Graphical, Analytical, and Numerical Approaches
There are two major types of discontinuities in functions.
Discontinuities
Removable
Non-removable
Point
Infinite
Jump
Description of the
Graph
Description of the
Graph
Description of the
Graph
The graph below has a
removable discontinuity at
x = 1.
The graph below has an
infinite non-removable
discontinuity at x = 1.
The graph below has a
jump non-removable
discontinuity at x = 1.
Domain:
Domain:
Domain:
Range:
Range:
Range:
2
For each of the functions graphed below, identify and classify any and all discontinuities that
exist. Then, state the domain and range of each function.
1.
Discontinuities
Domain_________________________
Range___________________________
2.
Discontinuities
Domain_________________________
Range___________________________
3.
Discontinuities
Domain_________________________
Range___________________________
3
The graph pictured below contains all three types of discontinuities. Identify the x values where
the function is discontinuous.
Removable Discontinuity
Jump Non-Removable
Discontinuity
Infinite Non-Removable
Discontinuity
Describe the y values of
the function as the x
values approach the x
value of discontinuity from
each side.
Describe the y values of
the function as the x
values approach the x
value of discontinuity from
each side.
Describe the y values of
the function as the x
values approach the x
value of discontinuity from
each side.
Complete the following statements that describe the y values of a function as x values approach
values of discontinuity.
1. As x values on either side of the discontinuity approach the value of a removable (point)
discontinuity, the y values approach ____________________ , provided the function value
does not equal the approached y – value.
2. As x values on either side of the discontinuity approach the value of a non-removable (jump)
discontinuity, the y values approach ________________________.
3. As x values on either side of the discontinuity approach the value of a non-removable
(infinite) discontinuity, the y values approach _______________ and/or _________________.
4
For each of the functions graphed below, describe the y values that the function approaches
from both sides of each discontinuity that exists in the function. The first function provides the
model of the statements you need to write for each function.
1.
The graph has a removable discontinuity at x = ____.

As x → ____ from the left, the y – values → ____.

As x → ____ from the right, the y – values →
____.
The graph has a non-removable (jump) discontinuity at x =
____.
2.
3.
5

As x → ____ from the left, the y – values →
____.

As x → ____ from the right, the y – values →
____.
f ( x) 
2 x2  5x  3
x2  x  6
If the coordinates of the hole in the graph are
3, 75  , complete each of the following
statements with a value.
As x→ −∞, the graph of 𝑓(𝑥 → _____.
As x → ∞, the graph of 𝑓(𝑥) → _____.
As x→3 from the left, the graph of 𝑓(𝑥) → ___.
As x→3 from the right, the graph of 𝑓(𝑥) → ___.
As x→ −2 from the left, the graph of 𝑓(𝑥) → __.
As x→−2 from the right, the graph of 𝑓(𝑥) → __.
Identify the domain and range of 𝑓(𝑥):
Domain:_____________________________
Range:______________________________
g ( x) 
If the coordinates of the hole in the graph are
 3, 12  , complete each of the following
x 2  4x  3
x 2  2x  3
statements with a value.
As x→ −∞, the graph of 𝑔(𝑥) → ____.
As x→ ∞, the graph of 𝑔(𝑥) → ____.
As x→ –3 from the left, the graph of 𝑔(𝑥) → __.
As x→–3 from the right, the graph of 𝑔(𝑥) → __.
As x→1 from the left, the graph of 𝑔(𝑥) → ____.
As x→1 from the right, the graph of 𝑔(𝑥) → ___.
Identify the domain and range of 𝑔(𝑥):
Domain:_____________________________
Range:______________________________
6
h( x ) 
x2
If the coordinates of the hole in the graph are
 2,1, complete the following statements with a
value.
x 2  5x  6
As x→ −∞, the graph of ℎ(𝑥) → ____.
As x → ∞, the graph of ℎ(𝑥) → ____.
As x→ −2 from the left, the graph of ℎ(𝑥) → __.
As x→−2 from the right, the graph of ℎ(𝑥) → __.
As x→−3 from the left, the graph of ℎ(𝑥) → ___.
As x→−3 from the right, the graph of ℎ(𝑥) → __.
Identify the domain and range of ℎ(𝑥):
Domain:_____________________________
Range:______________________________
p ( x) 
If the coordinates of the hole in the graph are
 2,1 , complete the following statements with
a value.
2x 2  7x  6
x 2  5x  6
As x → −∞, the graph of 𝑝(𝑥) → ____.
As x → ∞, the graph of 𝑝(𝑥) → ____.
As x→−2 from the left, the graph of 𝑝(𝑥) → ___.
As x→−2 from the right, the graph of 𝑝(𝑥) → __.
As x→−3 from the left, the graph of 𝑝(𝑥) → ___.
As x→−3 from the right, the graph of 𝑝(𝑥) → __.
Identify the domain and range of 𝑝(𝑥):
Domain:_____________________________
Range:______________________________
7
Based on your observations from the graphical analysis of the four functions on pages 5 and 6,
fill in the blanks in each statement below or answer the question asked.
1. If the graph of 𝑓(𝑥) → 𝑎 as 𝑥 → ∞ or −∞, then the graph of 𝑓(𝑥) has a
_________________________________________________________.
2. If the graph of 𝑓(𝑥) → 𝑏 as 𝑥 → 𝑎 from the left and the right, then the graph of 𝑓(𝑥) has a
_________________________________________________________.
3. If the graph of 𝑓(𝑥) → ∞ or −∞ as 𝑥 → 𝑎 from the left or the right, then the graph of 𝑓(𝑥)
has a
________________________________________________________.
4. How do you determine the domain of the rational functions you have seen so far?
5. How do you determine the range of the rational functions you have seen so far?
The table below shows selected values on the graph of a rational function, F(x). Use the
information in the table to answer the questions below.
x
F(x)
−500
1.499
−0.501
−248.5
−0.5
Undefined
−0.499
251.5
1.9
1.604
2
Undefined
2.1
1.596
500
1.501
(a) Identify the equation of any horizontal asymptote of the graph of F(x). Give a reason for
your answer.
(b) Identify the equation of any vertical asymptote of the graph of F(x). Give a reason for your
answer.
(c) Identify the coordinates of any hole in the graph of F(x). Give a reason for your answer.
(d) State the domain and range of the graph of F(x).
8
Connecting the Equation of a Rational Function to Its Graph
A Focus on Asymptotic Behavior
Non-cancelling Factors in the Denominator
Consider the function below and, based on the equation of the function, explain why the function
graphed below has a vertical asymptote at x = –3.
Equation of g(x):
g ( x) 
 2 x 2  5x  2
2x 2  7x  3
(2 x  1)( x  2)

(2 x  1)( x  3)
When the equation has a factor in the denominator that does not cancel out, then the graph has a(n)
____________________________ at the x value that makes that factor equal 0.
Cancelling Factors in the Numerator and Denominator
Consider the function below and, based on the equation of the function, explain why the graphed
function below has a hole in the graph at x =  1 . Also, find
2
the coordinates of the hole.
Equation of g(x):
g ( x) 
 2 x 2  5x  2
2x 2  7x  3
(2 x  1)( x  2)

(2 x  1)( x  3)
When the equation has a factor in the numerator and denominator that does cancel out, then the
graph has a(n) ____________________________ at the x value that makes that factor equal 0.
9
Non-cancelling Factors in the Numerator
One thing that we have not yet discovered is what happens graphically when there is a factor in the
numerator that does not cancel out.
Graph of f(x)
f ( x) 
Graph of g(x)
2 x2  7 x  6
g ( x) 
x2  5x  6
x 2  4x  3
x 2  2x  3
Rewrite the function in completely factored
form.
Rewrite the function in completely factored form.
What factor does not cancel out of the
numerator? If this factor is set equal to zero,
what is the value of x that results?
What factor does not cancel out of the
numerator? If this factor is set equal to zero,
what is the value of x that results?
By looking at the graphs above, what inference can you make about what appears graphically at x –
values that make the non-cancelling factors in the numerators equal zero?
When the equation has a factor in the numerator that does not cancel out, then the
graph has a(n) ____________________________ at the x value that makes that factor equal 0.
10
Shape Near the Origin
Graph 𝑓(𝑥) on your TI calculator.
𝑓(𝑥) = 𝑥 3 − 5𝑥
Describe the shape of the graph near the origin. (Hint: “Zoom In”
several times.)
Graph 𝑔(𝑥) on your calculator.
𝑔(𝑥) = 𝑥 3 − 5𝑥 2
Describe the shape of the graph near the origin. (Hint: “Zoom In”
several times.)
11
Existence of Horizontal Asymptotes – Comparing the Degrees of the Numerator &
Denominator
f ( x) 
x3
f ( x) 
2
x  x6
x2
2
x  5x  6
What do you notice about the degree of the numerator compared to the degree of the denominator for
the two f(x) functions above?
Where does each function have a horizontal asymptote?
x 2  3x  2
g ( x) 
x 1
x 2  5x  6
g ( x) 
2x  4
What do you notice about the degree of the numerator compared to the degree of the denominator for
the two g(x) functions above?
Where does each function have a horizontal asymptote?
12
h( x ) 
2x 2  7x  6
h( x ) 
x 2  5x  6
 2 x 2  5x  2
2x 2  7x  3
What do you notice about the degree of the numerator compared to the degree of the
denominator for the two h(x) functions above?
Where is the horizontal asymptote for the
function h( x) 
2
2x  7x  6
x 2  5x  6
?
a
, where a is the
b
leading coefficient of the numerator of h(x) and
b is the leading coefficient of the denominator
of h(x)?
What is the value of y 
Where is the horizontal asymptote for the
function h( x) 
 2 x 2  5x  2
2x 2  7x  3
?
a
, where a is the
b
leading coefficient of the numerator of h(x) and
b is the leading coefficient of the denominator
of h(x)?
What is the value of y 
Based on all that you have seen, make a list of three rules that govern the existence and
determination of the horizontal asymptote of a rational function.
1. ______________________________________________________________________________
______________________________________________________________________________
2. ______________________________________________________________________________
______________________________________________________________________________
3. ______________________________________________________________________________
______________________________________________________________________________
13
Existence of Slant Asymptotes
Notice that the graph below has a slant asymptote. This will always occur when the degree of the
numerator is exactly 1 greater than the degree of the denominator. Notice that the slant asymptotes
are oblique lines. Oblique lines are defined to be lines that are not vertical or horizontal. They have a
slope and a y – intercept.
From the graph, what is the equation of the
slant asymptote? Your equation should be in
the form of y = ax + b.
The equation of the graphed function to the
x 2  3x  2
right is g ( x) 
. Synthetically divide
x 1
the numerator by the denominator. What do
you notice?
From the graph, what is the equation of the
slant asymptote? Your equation should be in
the form of y = ax + b.
The equation of the graphed function to the
x2  x 1
. Synthetically divide the
2x  2
numerator by the denominator. What do you
notice?
right is h( x) 
14
For each of the following functions, determine if there exists a slant asymptote or not. Give a reason
for your answer. If one exists, determine the equation of it.
f ( x) 
 2x3  2x  2
x 1
1 x 2  5x  4
g ( x)  2
x4
15
g ( x) 
x 2  5x  6
2x  4
More Asymptotes
Ex. 1: Calculator NOT Permitted
Consider the rational function F ( x) 
2 x 2  5x  3
x2  x  6
to answer the following questions.
(a) If any exist, identify all equations of vertical asymptotes for 𝐹(𝑥) and explain how you know they
are vertical asymptotes.
(b) If any exist, identify all equations of horizontal asymptotes for 𝐹(𝑥) and explain how you know they
are horizontal asymptotes.
(c) If any exist, identify all coordinates of holes in the graph of 𝐹(𝑥) and explain how you know that
these are the coordinates of the holes in the graph.
(d) Sketch a graph of the function 𝐹(𝑥). Then, identify
the domain and range of 𝐹(𝑥).
16
Ex. 2: Determine the asymptotes for 𝑓(𝑥) =
𝑥 5 +2𝑥 4 +3
𝑥4
.
Ex. 3: Write a rational function satisfying the following conditions:
1. Vertical asymptote at 𝑥 = −1.
2. Horizontal asymptote at 𝑦 = 2.
3. Root of the function at 𝑥 = 3.
(Use of the grid is optional)
Ex. 4: Write a rational function satisfying the following conditions:
1. Vertical asymptote at 𝑥 = 2.
2. Slant asymptote at 𝑦 = 𝑥 + 1.
3. Root of the function at 𝑥 = −2.
(Use of the grid is optional)
17
Ex. 5: Do a complete analysis and graph the given function.
𝑥3
𝑓(𝑥) = 2
2𝑥 − 8
Symmetry
Intercepts
Asymptotes
𝑥-int.:____________
Vert.:_________________
𝑦-int.:____________
Horiz.:________________
Slant:_________________
End Behavior:
Excluded Region:
Hole:_________________
Extent:
Domain:____________________________
Range:_____________________________
18
Ex. 6: Do a complete analysis and graph the given function.
𝑦=
Symmetry
𝑥−1
𝑥2 − 4
Intercepts
Asymptotes
𝑥-int.:____________
Vert.:_________________
𝑦-int.:____________
Horiz.:________________
Slant:_________________
End Behavior:
Excluded Region:
Hole:_________________
Extent:
Domain:____________________________
Range:_____________________________
19
Complete Analysis
Complete Analysis - Symmetry, Intercepts, Asymptotes, Excluded Region, End Behavior, Domain
and Range, Graph (including Shading, Asymptotes, Plot Points)
Ex. 1: Do a complete analysis and graph the given function.
𝑓(𝑥) =
Symmetry
𝑥−1
𝑥2 − 4
Intercepts
Asymptotes
𝑥-int.:____________
Vert.:_________________
𝑦-int.:____________
Horiz.:________________
Slant:_________________
End Behavior:
Excluded Region:
Hole:_________________
Extent:
Domain:____________________________
Range:_____________________________
20
Ex. 2: Do a complete analysis and graph the given function.
𝑥2 − 𝑥 − 2
𝑓(𝑥) =
𝑥−1
Symmetry
Intercepts
Asymptotes
𝑥-int.:____________
Vert.:_________________
𝑦-int.:____________
Horiz.:________________
Slant:_________________
End Behavior:
Excluded Region:
Hole:_________________
Extent:
Domain:____________________________
Range:_____________________________
21
Ex. 3: Do a complete analysis and graph the given function.
3𝑥 4
𝑓(𝑥) = 4
𝑥 −1
Symmetry
Intercepts
Asymptotes
𝑥-int.:____________
Vert.:_________________
𝑦-int.:____________
Horiz.:________________
Slant:_________________
End Behavior:
Excluded Region:
Hole:_________________
Extent:
Domain:____________________________
Range:_____________________________
22
Ex. 4: Do a complete analysis and graph the given function.
𝑓(𝑥) =
Symmetry
(𝑥 − 7)(𝑥 − 1)
𝑥(𝑥 − 5)(𝑥 − 2)2
Intercepts
Asymptotes
𝑥-int.:____________
Vert.:_________________
𝑦-int.:____________
Horiz.:________________
Slant:_________________
End Behavior:
Excluded Region:
Hole:_________________
Extent:
Domain:____________________________
Range:_____________________________
23
Analysis of Rational Functions
Analytical, Graphical, and Numerical Approaches
For questions 1 – 6, refer to the rational function F ( x) 
(2 x  3)( x  2)
.
( x  3)( x  2)
1. What is the domain of the function 𝐹(𝑥)?
2. Does the graph of 𝐹(𝑥) have any holes in it? Why or why not? If any holes exist, what are the
coordinates of the holes?
3. Does the graph of 𝐹(𝑥) have any vertical asymptotes? Why or why not? If any vertical
asymptotes exist, what is/are the equations?
4. Does the graph of 𝐹(𝑥) have any horizontal asymptotes? Why or why not? If any horizontal
asymptotes exist, what is/are the equations?
5. The graph of 𝐹(𝑥) does not have a slant asymptote. Explain why.
6. What is the range of the function 𝐹(𝑥).
7. Sketch the graph of 𝐹(𝑥) on the axes to the right.
24
The table below shows function values for a rational function, 𝐺(𝑥). The equation of 𝐺(𝑥) is such that
(𝑥 + 2) and (𝑥 – 1) are the only factors in the denominator of the function.
𝒙
–1000
–2.001
–2
–1.999
0.999
1
1.001
1000
𝑮(𝒙)
0.998
0.333
Undefined
0.333
–1999
Undefined
2001
1.002
8. What is the domain of 𝐺(𝑥)?
9. Does either factor in the denominator also exist in the numerator? Give a reason for your answer.
10. Does either factor of the denominator not exist in the numerator? Give a reason for your answer.
11. Based on the end behavior, where does 𝐺(𝑥) have a horizontal asymptote? Give a reason for
your answer.
12. Sketch a possible graph of the function 𝐺(𝑥).
25
The table below shows values for a rational function, 𝐻(𝑥). Determine if the statements that follow
are True or False. Give a reason for your answer.
The denominator of 𝐻(𝑥) is 𝑥2 – 5𝑥 + 4.
13. The factor (𝑥 – 1) is a factor in both the numerator and denominator in the equation of 𝐻(𝑥).
14. The factor (𝑥 – 4) is a non-canceling factor in the denominator of the equation of 𝐻(𝑥).
15. As 𝑥 → ∞, then the graph of 𝐻(𝑥) → 2.
16. The graph of 𝐻(𝑥) has a horizontal asymptote
at 𝑦 = 2.
17. Sketch a possible graph of
𝐻(𝑥) on the grid to the right.
26
RATIONAL FUNCTIONS: __________________________________
27
Pictured below is the graph of a rational function, 𝐹(𝑥). Use the graph to answer the questions that
follow. The coordinates of the hole in the graph are 3, 4 5 .


18. What factor is guaranteed to be in both the numerator
and the denominator? Explain your reasoning.
19. What can be said about the degree of the numerator
compared to the degree of the denominator? Explain
your reasoning.
20. What factor is in the numerator that does NOT cancel out? Explain your reasoning.
21. What factor is in the denominator that does NOT cancel out? Explain your reasoning.
23. State the domain and range of 𝐹(𝑥).
22. Give a possible equation for the
function 𝐹(𝑥).
28
Pictured below is the graph of 𝐺(𝑥), the graph of a rational function. The coordinates of the hole in
the graph are  2, 1 . Determine if the following statements are True or False. Give a reason for

3

your choice.
24. The domain of the function is  ,1  1,  .
25. The degree of the numerator is less than the degree
the denominator.
26. As 𝑥 → ∞, then the graph of 𝐺(𝑥) → 1.
27. The factor (𝑥 + 1) is a non-canceling factor in the numerator of the equation of 𝐺(𝑥).
28. The factor (𝑥 – 1) is a non-canceling factor in the denominator of the equation of 𝐺(𝑥).

 3 
29. The range of the function is  , 1  1 ,1  1,  .
3
30. The factor (𝑥 – 2) is a factor in the numerator and denominator of the function.
31. The value of 𝐺(0) is –1.
29
of
The graph of a rational function, 𝐻(𝑥) is pictured to
the right. Use the graph to answer the following
questions.
32. What are the domain and range of 𝐻(𝑥)?
33. What factor is in the denominator that is not
also in the numerator? Explain your
reasoning.
34. What factor is in the denominator that is also
in the numerator? Explain your reasoning.
35. What factor is in the numerator that is NOT in the denominator? Explain your reasoning.
36. If a is the leading coefficient of the numerator and b is the leading coefficient of the denominator,
𝑎
then what is the value of ? Explain your reasoning.
𝑏
37. What is the equation of the function, 𝐻(𝑥)?
30
Review For Test on Graphing Techniques
Topics:
a) Complete Analysis of a Fucntion: Symmetry, Intercepts, Asymptotes, Excluded Region, End
Behavior, Domain/Range, Graph (Shade, Asymptotes, Plot Points)
b)
c)
d)
e)
Symmetry: Odd/Even
Transformations Rules and of Functions (Translated Form, completing the square)
Rotation of Axes Formula
Continuity
Part I: Complete Analysis (40pts)
Part II: Answer 15 out of 20 (60pts)
Examples:
1. How can you graph y  10 x  2  5 by using the graph of y  10 x
2. For each function below determine if it is continuous. If the function is not continuous at a
point, find the point that would make the function continuous.
a) y 
x 2  25
x5
b) f ( x) 
x2  x
x 1
3. The graph of y  4 x 2  16 x  5 is the same as y  4x 2 only if the origin is shifted to what
point?
31