Chapter 1 Functions, Graphs, and Limits · 2019. 8. 27. · Chapter 1 Functions, Graphs, and Limits...
Transcript of Chapter 1 Functions, Graphs, and Limits · 2019. 8. 27. · Chapter 1 Functions, Graphs, and Limits...
Chapter 1Functions, Graphs, and Limits
MA1103 Business Mathematics I
Semester I Year 2019/2020
SBM International Class K-10
Lecturer: Dr. Rinovia Simanjuntak
1.1 Functions
2
Function
3
A function is a rule that assigns to each object in a set 𝐴 exactly one object in a set 𝐵.
The set 𝐴 is called the domain of the function, and the set of assigned objects in 𝐵 is called the range.
Which One is a Function?
fff
A B
AB
f
A B f
4
)(xfy =
5
We represent a functional relationship by an equation
x and y are called variables: y is the dependent variableand x is the independent variable.
Example.
Note that x and y can be substituted by other letters. For example, the above function can be represented by
4)( 2 +== xxfy
42 += ts
Function which is Described as a Tabular Data
Academic Year Tuition and
Ending in Period n Fees
1973 1 $1,898
1978 2 $2,700
1983 3 $4,639
1988 4 $7,048
1993 5 $10,448
1998 6 $13,785
2003 7 $18,273
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Table 1.1 Average Tuition and Fees for 4-Year Private Colleges
=
periodyear -5th theof beginning
at the fees and tuition average)(
nnf
7
We can describe this data as a function f defined by the rule
Thus,
Noted that the domain of f is the set of integers
273,18)7(,,700,2)2(,898,1)1( === fff
}7,....,2,1{=A
Piecewise-defined function
8
A piecewise-defined function is a function that is often defined using more than one formula, where each individual formula describes the function on a subset of the domain.
Example.
Find 𝑓(−1/2), 𝑓(1), and 𝑓(2).
+
−=
1 xif 13
1 xif 1
1
)(2x
xxf
Natural Domain
The natural domain of 𝑓 is the domain of 𝑓 to be the set of all real numbers for which 𝑓(𝑥) is defined.
Examples.
Find the domain and range of each of these functions.
1.
2. 9
There are two situations often need to be considered:1) division by 02) the even root of a negative number
21
1)(
xxf
−=
4 2)( += uug
Functions Used in Economics
A demand function 𝑝 = 𝐷(𝑥) is a function that relates the unit price 𝑝 for a commodity to the number of units 𝑥demanded by consumers at that price.
The total revenue is given by the product𝑅 𝑥 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑡𝑒𝑚𝑠 𝑠𝑜𝑙𝑑 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑖𝑡𝑒𝑚
= 𝑥𝑝 = 𝑥𝐷(𝑥)
If 𝐶(𝑥) is the total cost of producing the x units, then the profit is given by the function
𝑃(𝑥) = 𝑅(𝑥) − 𝐶(𝑥) = 𝑥𝐷(𝑥) − 𝐶(𝑥)
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ExampleMarket research indicates that consumers will buy 𝑥thousand units of a particular kind of coffee maker when the unit price is 𝑝 = −0.27𝑥 + 51 dollars. The cost of producing the 𝑥 thousand units is
𝐶 𝑥 = 2.23 𝑥2 + 3.5𝑥 + 85
855.323.2)( 2 ++= xxxC
thousand dollars
a. What are the revenue and profit functions, 𝑅(𝑥) and 𝑃(𝑥), for this production process?
b. For what values of 𝑥 is production of the coffee makers profitable?
Composition of Functions
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Given functions 𝑓(𝑢) and 𝑔(𝑥), the composition 𝑓(𝑔(𝑥)) is the function of 𝑥 formed by substituting 𝑢 = 𝑔(𝑥) for 𝑢 in the formula for 𝑓(𝑢).
Example.
Find the composition function 𝑓(𝑔(𝑥)), where 𝑓 𝑢 = 𝑢3 + 1and 𝑔 𝑥 = 𝑥 + 1.
Question: How about 𝑔(𝑓(𝑥))?
Note: In general, 𝑓(𝑔(𝑥)) and 𝑔(𝑓(𝑥)) are not the same.
1.2 The Graph of a Function
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GraphThe graph of a function 𝑓 consists of all points (𝑥, 𝑦) where 𝑥 is in the domain of 𝑓 and 𝑦 = 𝑓(𝑥), that is, all points of the form (𝑥, 𝑓(𝑥)).
Rectangular coordinate system, horizontal axis, vertical axis.
2)( 2 ++−= xxxf
x -3 -2 -1 0 1 2 3 4
f(x) -10 -4 0 2 2 0 -4 -10
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Intercepts
𝑥 intercept: points where a graph crosses the x axis.
𝑦 intercept: a point where the graph crosses the y axis.
How to find the 𝒙 and 𝒚 intercepts:
• The only possible 𝑦 intercept for a function is 𝑦0 = 𝑓(0).
• To find any 𝑥 intercept of 𝑦 = 𝑓(𝑥), set 𝑦 = 0 and solve for 𝑥.
Note: Sometimes finding 𝑥 intercepts may be difficult.
In the aforementioned example, the 𝑦 intercept is 𝑓(0) = 2. To find the 𝑥 intercepts, solve the equation 𝑓(𝑥) = 0, we have 𝑥 = −1 and 2. Thus, the 𝑥 intercepts are (−1,0) and (2,0). 15
Parabolas
Parabolas: The graph of 𝑦 = 𝐴𝑥2 + 𝐵𝑥 + 𝐶, 𝐴 ≠ 0.
All parabolas have a “U shape” and the parabola opens up if 𝐴 > 0 and down if 𝐴 < 0.
The “peak” or “valley” of the parabola is called its vertex,
and it always occurs where 𝑥 =−𝐵
2𝐴.
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Example
A manufacturer determines that when 𝑥 hundred units of a particular commodity are produced, they can all be sold for a unit price given by the demand function 𝑝 = 60 − 𝑥 dollars. At what level of production is revenue maximized? What is the maximum revenue?
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Intersections of Graphs
Sometimes it is necessary to determine when two functions are equal.
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For example, an economist may wish to compute the market price at which the consumer demand for a commodity will be equal to supply.
Power, Polynomial, and Rational Functions
A power function: A function of the form 𝑓 𝑥 = 𝑥𝑛, where 𝑛 is a real number.
A polynomial function: A function of the form 𝑝 𝑥 = 𝑎𝑛𝑥
𝑛 + 𝑎𝑛−1𝑥𝑛−1 +⋯+ 𝑎1𝑥 + 𝑎0
where 𝑛 is a nonnegative integer and 𝑎0, 𝑎1, … , 𝑎𝑛 are constants.
If 𝑎𝑛 ≠ 0, the integer 𝑛 is called the degree of the polynomial.
A rational function: A quotient 𝑝(𝑥)
𝑞(𝑥)of two polynomials 𝑝(𝑥) and 𝑞(𝑥).
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The Vertical Line Test
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A curve is the graph of a function if and only if no vertical line intersects the curve more than once.
1.3 Linear Functions
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Linear Functions
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A linear function is a function that changes at a constant rate with respect to its independent variable.
The graph of a linear function is a straight line.
The equation of a linear function can be written in the form
𝑦 = 𝑚𝑥 + 𝑏
where 𝑚 and 𝑏 are constants.
The Slope of a Line
The slope of the non-vertical line passing through the points (𝑥1, 𝑦1) and (𝑥2, 𝑦2) is given by the formula
slope =change in 𝑦
change in 𝑥=∆𝑦
∆𝑥=𝑦2 − 𝑦1𝑥2 − 𝑥1
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Equation of a Line The slope-intercept form: The equation 𝑦 = 𝑚𝑥 + 𝑏 is the equation of a line whose slope is 𝑚 and whose 𝑦 intercept is (0, 𝑏).
The point-slope form: The equation 𝑦 − 𝑦0 = 𝑚(𝑥 − 𝑥0)is an equation of the line that passes through the point (𝑥0, 𝑦0) and has slope equal to 𝑚.
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The slope-intercept form is 3
1
)05.1(
)5.00(−=
−−
+=m
2
1
3
1−−= xy
The point-slope form that passes through the point (−1.5,0) is
)5.1(3
10 +−=− xy
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Table 1.2 lists the percentage of the labour force that was unemployed during the decade 1991-2000. Plot a graph with the time (years after 1991) on the x axis and percentage of unemployment on the y axis. Do the points follow a clear pattern? Based on these data, what would you expect the percentage of unemployment to be in the year 2005?
Number of Years Percentage of
Year from 1991 Unemployed
1991 0 6.8
1992 1 7.5
1993 2 6.9
1994 3 6.1
1995 4 5.6
1996 5 5.4
1997 6 4.9
1998 7 4.5
1999 8 4.2
2000 9 4.0
Table 1.2 Percentage of Civilian Unemployment
Parallel and Perpendicular Lines
Let and be the slope of the non-vertical lines
and . Then
and are parallel if and only if
and are perpendicular if and only if
1m
2L
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2m1L
1L 2L 21 mm =
1L 2L1
2
1
mm −=
27
Let L be the line 4x+3y=3
a. Find the equation of a line parallel to L through P(-1,4).
b. Find the equation of a line perpendicular to L through Q(2,-3).
1L
2L
Solution:
By rewriting the equation 4x+3y=3 in the slope-intercept form
, we see that L has slope
a. Any line parallel to L must also have slope -4/3. The required line
contains P(-1,4), we have
b. A line perpendicular to L must have slope m=3/4. Since the
required line contains Q(2,-3), we have
13
4+−= xy
3
4−=Lm
1L3
8
3
4)1(
3
44 +−=+−=− xyxy
2
9
4
3
)2(4
33
−=
−=+
xy
xy2L
1.4 Functional Models
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Functional Models
To analyze a real world problem, a common procedure is to make assumptions about the problem that simplify it enough to allow a mathematical description. This process is called mathematical modelling and the modified problem based on the simplifying assumptions is called a mathematical model.
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Real-world
problem
Testing
Interpretation
Mathematical
model
adjustments
PredictionAnalysis
Formulation
Elimination of Variables
In next example, the quantity you are seeking is expressed most naturally in term of two variables. We will have to eliminate one of these variables before you can write the quantity as a function of a single variable.
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Example
The highway department is planning to build a picnic area for motorists along a major highway. It is to be rectangular with an area of 5,000 square yards and is to be fenced off on the three sides not adjacent to the highway. Express the number of yards of fencing required as a function of the length of the unfenced side.
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Modelling in Business and Economics
A manufacturer can produce blank videotapes at a cost of $2 per cassette. The cassettes have been selling for $5 a piece. Consumers have been buying 4000 cassettes a month. The manufacturer is planning to raise the price of the cassettes and estimates that for each $1 increase in the price, 400 fewer cassettes will be sold each month.
a. Express the manufacturer’s monthly profit as a function of the price at which the cassettes are sold.
b. Sketch the graph of the profit function. What price corresponds to maximum profit? What is the maximum profit?
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Market Equilibrium
The law of supply and demand: In a competitive market environment, supply tends to equal demand, and when this occurs, the market is said to be in equilibrium.
The demand function:
𝑝 = 𝐷(𝑥)
The supply function:
𝑝 = 𝑆(𝑥)
The equilibrium price:
𝑝𝑒 = 𝐷 𝑥𝑒 = 𝑆(𝑥𝑒)
Shortage: 𝐷(𝑥) > 𝑆(𝑥)
Surplus: 𝑆(𝑥) > 𝐷(𝑥)
Example
Market research indicates that manufacturers will supply 𝑥units of a particular commodity to the marketplace when the price is 𝑝 = 𝑆(𝑥) dollars per unit and that the same number of units will be demanded by consumers when the price is 𝑝 =𝐷(𝑥) dollars per unit, where the supply and demand functions are given by
𝑆 𝑥 = 𝑥2 + 14 𝐷 𝑥 = 174 − 6𝑥
a. At what level of production 𝑥 and unit price 𝑝 is market equilibrium achieved?
b. Sketch the supply and demand curves, 𝑝 = 𝑆(𝑥) and 𝑝 =𝐷(𝑥), on the same graph and interpret.
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Break-Even Analysis
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At low levels of production, the manufacturer suffers a
loss. At higher levels of production, however, the total
revenue curve is the higher one and the manufacturer
realizes a profit.
Break-even point: The total revenue equals total cost.
Example
A manufacturer can sell a certain product for $110 per unit. Total cost consists of a fixed overhead of $7500plus production costs of $60 per unit.
a. How many units must the manufacturer sell to break even?
b. What is the manufacturer’s profit or loss if 100units are sold?
c. How many units must be sold for the manufacturer to realize a profit of $1250?
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Example (2)
A certain car rental agency charges $25 plus 60cents per mile. A second agency charge $30plus 50 cents per mile.
Which agency offers the better deal?
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1.5 Limits
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Illustration of Limit
The limit process involves examining the behaviour of a function f(x) as x approaches a number c that may or may not be in the domain of f.
Illustration.
Consider a manager who determines that when x percent of her company’s plant capacity is being used, the total cost is
hundred thousand dollars. The company has a policy of rotating maintenance in such a way that no more than 80% of capacity is ever in use at any one time. What cost should the manager expect when the plant is operating at full permissible capacity? 41
96068
3206368)(
2
2
−−
−−=
xx
xxxC
42
It may seem that we can answer this question by simply
evaluating C(80), but attempting this evaluation results in
the meaningless fraction 0/0.
However, it is still possible to evaluate C(x) for values of
x that approach 80 from the left (x<80) and the right
(x>80), as indicated in this table:
x approaches 80 from the left → ←x approaches 80 from the right
x 79.8 79.99 79.999 80 80.0001 80.001 80.04
C(x) 6.99782 6.99989 6.99999 7.000001 7.00001 7.00043
The values of C(x) displayed on the lower line of this table
suggest that C(x) approaches the number 7 as x gets closer
and closer to 80. The functional behavior in this example
can be describe by lim𝑥→80 𝐶 𝑥 = 7.
Limits
If 𝑓(𝑥) gets closer and closer to a number 𝐿 as 𝑥 gets closer and closer to c from both sides, then 𝐿 is the limit of 𝑓(𝑥) as 𝑥 approaches 𝑐. The behaviour is expressed by writing
lim𝑥→𝑐
𝑓(𝑥) = 𝐿
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Example
Use a table to estimate the limit
44
1
1)(
−
−=
x
xxfLet and compute f(x) for a succession of values
of x approaching 1 from the left and from the right.
1
1lim
1 −
−
→ x
x
x
x→ 1 ← x
x 0.99 0.999 0.9999 1 1.00001 1.0001 1.001
f(x) 0.50126 0.50013 0.50001 0.499999 0.49999 0.49988
The table suggest that f(x) approaches 0.5 as x approaches
1. That is 5.0
1
1lim
1=
−
−
→ x
x
x
45
Three functions for which
It is important to remember that limits describe the behavior of a function near a particular point, not necessarily at the point itself.
4)(lim3
=→
xfx
46
The figure below shows that the graph of two functions
that do not have a limit as x approaches 2.
Figure (a): The limit does not exist;
Figure (b): The function has no finite limit as x
approaches 2. Such so-called infinite limits will be
discussed later.
Properties of Limits
thenexist, )(limand)(lim If xgxfcxcx →→
)(lim)(lim)]()([lim xgxfxgxfcxcxcx →→→
+=+
)(lim)(lim)]()([lim xgxfxgxfcxcxcx →→→
−=−
constant any for )(lim)(lim kxfkxkfcxcx →→
=
47
)](lim)][(lim[)]()([lim xgxfxgxfcxcxcx →→→
=
0)(lim if )(lim
)(lim]
)(
)([lim =
→
→
→
→xg
xg
xf
xg
xf
cx
cx
cx
cx
exists )](lim[ if )](lim[)]([lim p
cx
p
cx
p
cxxfxfxf
→→→=
48
For any constant k,
That is, the limit of a constant is the constant itself, and
the limit of f(x)=x as x approaches c is c.
cxkkcx
==→→ cx
lim and lim
Examples
)843(lim 3
1+−
−→xx
x 2
83lim
3
0 −
−
→ x
x
x
49
Find (a) (b)
Limits of Polynomials and Rational Functions
50
If p(x) and q(x) are polynomials, then
and
)()(lim cpxpcx
=→
0)( if )(
)(
)(
)(lim =→
cqcq
cp
xq
xp
cx
Example.
Find 2
1lim
2 −
+
→ x
x
x
The quotient rule for limits does not apply in this case since the limit of
the denominator is 0 and the limit of the numerator is 3.
Indeterminate Form
51
If and , then is said to be
indeterminate. The term indeterminate is used since the limit
may or may not exist.
Examples.
(a) Find (b) Find
0)(lim =→
xfcx
0)(lim =→
xgcx )(
)(lim
xg
xf
cx→
23
1lim
2
2
1 +−
−
→ xx
x
x 1
1lim
1 −
−
→ x
x
x
Limits Involving Infinity
52
Limits at Infinity
If the value of the function f(x) approach the number L as x increases
without bound, we write
Similarly, we write
when the functional values f(x) approach the number M as x decreases
without bound.
Lxfx
=+→
)(lim
Mxfx
=−→
)(lim
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Reciprocal Power Rules
For constants A and k, with k>0, 0lim and 0lim ==−→+→ kxkx x
A
x
A
Example.
Find2
2
21lim
xx
x
x +++→
5.0200
1
2lim/1lim/1lim
1lim
/2//1
/lim
21lim
22222
22
2
2
=++
=++
=++
=++
+→+→+→
+→
+→+→
xxx
x
xx xxxxxxx
xx
xx
x
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Procedure for Evaluating a Limit at Infinity of f(x)=p(x)/q(x)
Step 1. Divide each term in f(x) by the highest power xk that
appears in the denominator polynomial q(x).
Step 2. Compute or using algebraic
properties of limits and the reciprocal rules.
)(lim xfx +→
)(lim xfx −→
Example.
15
283lim
4
24
+
+−
+→ x
xxx
x
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Infinite Limits
If f(x) increases or decreases without bound as x→c, we
have )(limor )(lim xf xfcxcx
−=+=→→
Example. 22 )2(lim
−→ x
x
x
From the figure, we
can guest that
+=−→ 22 )2(
limx
x
x
1.6 One-sided Limits and Continuity
56
One-Sided Limits
If f(x) approaches L as x tends toward c from the left (x<c), we write
Lxfcx
=−→
)(lim
Mxfcx
=+→
)(lim
57
where L is called the limit from the left (or left-hand
limit)
Likewise if f(x) approaches M as x tends toward c
from the right (x>c), then
M is called the limit from the right (or right-hand
limit).
Example.
For the function
evaluate the one-sided limits and
58
+
−=
2 if 12
2 if 1)(
2
xx
xxxf
)(lim2
xfx −→
)(lim2
xfx +→
Since for x<2, we have 21)( xxf −=
3)1(lim)(lim 2
22−=−=
−− →→xxf
xx
Similarly, f(x)=2x+1 if x≥2, so
5)12(lim)(lim22
=+=++ →→
xxfxx
59
Existence of a Limit
The two-sided limit exists if and only if the two
one-sided limits and exist and are
equal, and then
)(lim xfcx→
)(lim xfcx −→
)(lim xfcx +→
)(lim)(lim)(lim xfxfxfcxcxcx +− →→→
==
Recall.
Find 2
1lim
2 −
+
→ x
x
x
60
At x=1: ( )1
lim 0x
f x−→
=
( )1
lim 1x
f x+→
=
( )1 1f =
Left-hand limit
Right-hand limit
value of the function
does not exist!
Since the left and right hand
limits are not equal.
)(lim1
xfx→
61
At x=2: Left-hand limit
Right-hand limit
value of the function
( )2
lim 1x
f x−→
=
( )2
lim 1x
f x+→
=
( )2 2f =
does exist!
Since the left and right
hand limits are equal.
However, the limit is not
equal to the value of
function.
)(lim2
xfx→
62
At x=3: Left-hand limit
Right-hand limit
value of the function
( )3
lim 2x
f x−→
=
( )3
lim 2x
f x+→
=
( )3 2f =
does exist!
Since the left and right
hand limits are equal,
and the limit is equal
to the value of
function.
)(lim3
xfx→
63
Non-existent One-sided Limits
A simple example is provided by the function
)/1sin()( xxf =
As x approaches 0 from either the left or the right, f(x)
oscillates between -1 and 1 infinitely often. Thus neither
one-sided limit at 0 exists.
64
Continuity
A continuous function is one whose graph can be drawn
without the “pen” leaving the paper. (no holes or gaps )
65
A “hole “ at x=c
66
A “gap” at x=c
67
What properties will guarantee that f(x) does not have a “hole”
or “gap” at x=c?
A function f is continuous at c if all three of these conditions
are satisfied:
a.
b.
c.
If f(x) is not continuous at c, it is said to have a discontinuity
there.
exists )(lim xfcx→
)()(lim cfxfcx
=→
defined is )(cf
68
f(x) is continuous at
x=3 because the left
and right hand limits
exist and equal to f(3).
At x=1:
At x=2:
At x=3:
)(lim)(lim11
xfxfxx −+ →→
)2()(lim)(lim22
fxfxfxx
=−+ →→
)3()(lim)(lim33
fxfxfxx
==−+ →→
Discontinuous
Discontinuous
Continuous
69
Continuity of Polynomials and
Rational Functions
If p(x) and q(x) are polynomials, then
)()(lim cpxpcx
=→
0)( if )(
)(
)(
)(lim =→
cqcq
cp
xq
xp
cx
A polynomial or a rational function is continuous
wherever it is defined
Example
1. Show that the rational function 𝑓 𝑥 =𝑥+1
𝑥−2is
continuous at 𝑥 = 3.
2. Determine where the function below is not continuous.
ℎ 𝑡 =4𝑡 + 10
𝑡2 − 2𝑡 − 15
70
Example (2)
Discuss the continuity of each of the following functions
71
−
+=
+
−==
1 if 2
1 if 1)( .
1
1)( .
1)( .
2
xx
xxxhc
x
xxgb
xxfa
Example (3)
1. For what value of the constant A is the following function continuous for all real x?
2. Find numbers a and b so that the following function is continuous everywhere.
72
+−
+=
1 xif 43
1 if 5)(
2 xx
xAxxf
−−+
=
1 if
11 if
1 if
)( 2
xbx
xbax
-xax
xf
Continuity on an Interval
A function 𝑓(𝑥) is said to be continuous on an open interval𝑎 < 𝑥 < 𝑏 if it is continuous at each point 𝑥 = 𝑐 in that interval.
f is continuous on closed interval 𝑎 ≤ 𝑥 ≤ 𝑏, if it continuous on the open interval 𝑎 < 𝑥 < 𝑏, lim𝑥→𝑎+ 𝑓(𝑥) = 𝑓(𝑎) and lim𝑥→𝑏−
𝑓(𝑥) = 𝑓 𝑏 .
Example.
1. 𝑓 𝑥 = 1 − 𝑥2
2. Discuss the continuity of the function 𝑔 𝑥 =𝑥+2
𝑥−3on the open interval −2 < 𝑥 < 3 and on the closed interval −2 ≤ 𝑥 ≤ 3.
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