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An in-depth exploration of intervals, their notation, set descriptions, and graphical representations. It covers various types of intervals, including finite and infinite intervals, and discusses their relationship with functions and their graphs. The document also introduces transformations of functions, such as shifting, reflecting, and stretching, and demonstrates how to apply these transformations to specific examples.
Typology: Summaries
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Intervals
Certain sets of real numbers, called intervals, occur frequently in calculus and correspond geometrically to line segments. For example, if a < b, the open interval from a to b consists of all numbers between a and b and is denoted by the symbol (a, b). Using set-builder notation, we can write
(a, b) = {x ∈ R
a < x < b}
Intervals
(a, b) = {x|a < x < b}
Notice that the endpoints of the interval - namely, a and b - are excluded. This is indicated by the round brackets ( ) and by the open dots in Figure.
Intervals, Inequalities, and Abso
Certain sets of real numb spond geometrically to l to consists of all num Using set-builder notatio
Notice that the endpoin indicated by the round br val from to is the set
Here the endpoints of th ets and by the solid do in an interval, as shown We also need to cons
a b
a b
A2 APPENDIX A INTERVALS, INEQUALITIES, AND ABSOLUTE VAL
a b FIGURE 1 Open interval (a, b)
a b FIGURE 2 Closed interval [a, b]
Intervals
It is also possible to include only one endpoint in an interval, as shown in Table 1. We also need to consider infinite intervals such as
(a, ∞) = {x ∈ R
∣ (^) x > a}
This does not mean that ∞ (“infinity”) is a number. The notation (a, ∞) stands for the set of all numbers that are greater than a, so the symbol ∞ simply indicates that the interval extends indefinitely far in the positive direction.
Intervals
Notation Set description Picture (a, b) {x : a < x < b}
Notation Set description Picture Notation
Table of Intervals
Inequalities When working with inequalities, not
Rules for Inequalities
1. If , then. 2. If a band c d, then a
a b a c b c
1
,
, b
, b
a,
a b^ a,^
a b
a b
a b
[a, b] {x : a ≤ x ≤ b}
Notation Set description Picture Notation
Table of Intervals
Inequalities When working with inequalities, not
Rules for Inequalities
1. If , then. 2. If and , then 3. If a band c 0 , then ac
a b c d a
a b a c b c
1
,
, b
, b
a,
a b^ a,^
a b
a b
a b
[a, b) {x : a ≤ x < b}
Notation Set description Picture Notation
Table of Intervals
Inequalities When working with inequalities, not
Rules for Inequalities
1. If , then. 2. If and , then 3. If a band c 0 , then ac
a b c d a
a b a c b c
1
,
, b
, b
a,
a b^ a,^
a b
a b
a b
(a, b] {x : a < x ≤ b}
Notation Set description Picture Notation
Inequalities When working with inequalities, not
Rules for Inequalities
1. If , then. 2. If and , then 3. If a band c 0 , then ac
a b c d a
a b a c b c
1
,
, b
, b
a,
a b^ a,^
a b
a b
a b (a, ∞) {x : x > a}
ription Picture
lso need to consider infinite intervals such as
s not mean that (“infinity”) is a number. The notation stands for the l numbers that are greater than , so the symbol simply indicates that the extends indefinitely far in the positive direction.
equalities orking with inequalities, note the following rules.
a
a,
a, x (^) x a
(set of all real numbers)
,
a a b b
[a, ∞) {x : x ≥ a}
ription Picture
lso need to consider infinite intervals such as
s not mean that (“infinity”) is a number. The notation stands for the l numbers that are greater than , so the symbol simply indicates that the extends indefinitely far in the positive direction.
equalities
a
a,
a, x (^) x a
(set of all real numbers)
,
a a b b
(−∞, b) {x : x < b}
ription Picture
lso need to consider infinite intervals such as
s not mean that (“infinity”) is a number. The notation stands for the l numbers that are greater than , so the symbol simply indicates that the extends indefinitely far in the positive direction.
equalities
a
a,
a, x (^) x a
(set of all real numbers)
,
a a b (−∞, b] {x : x ≤ b} b
ription Picture
lso need to consider infinite intervals such as
s not mean that (“infinity”) is a number. The notation stands for the l numbers that are greater than , so the symbol simply indicates that the extends indefinitely far in the positive direction.
equalities
a
a,
a, x (^) x a
(set of all real numbers)
,
a a b b (−∞, ∞) R
ription Picture
lso need to consider infinite intervals such as
s not mean that (“infinity”) is a number. The notation stands for the l numbers that are greater than , so the symbol simply indicates that the extends indefinitely far in the positive direction.
equalities
a
a,
a, x (^) x a
(set of all real numbers)
,
a a b b
Tablo 1: Table of intervals
MAT 1001 Calculus I 5 / 79
Inequalities
Solve the inequality x^2 − 5 x + 6 ≤ 0
First we factor the left side:
(x − 2)(x − 3) ≤ 0
We know that the corresponding equation (x − 2)(x − 3) = 0 has the solutions 2 and 3. The numbers 2 and 3 divide the real line into three intervals: (−∞, 2) (2, 3) (3, ∞)
Inequalities
On each of these intervals we determine the signs of the factors.
x (x − 2) (x − 3) (x − 2)(x − 3)
Then we read from the chart that (x − 2)(x − 3) is negative when 2 < x < 3. Thus, the solution of the inequality (x − 2)(x − 3) ≤ 0 is
{x ∈ R
2 ≤ x ≤ 3 } = [2, 3]
Notice that we have included the endpoints 2 and 3 because we are looking for values of such x that the product is either negative or zero.
Inequalities
On each of these intervals we determine the signs of the factors.
x
x
(x − 1)
(x + 4)
x(x − 1)(x + 4)
Then we read from the chart that the solution set is
{x ∈ R
− 4 < x < 0 or x > 1 } = (− 4 , 0) ∪ (1, ∞).
Four Ways to Represent a Function
Four Ways to Represent a Function
Functions arise whenever one quantity depends on another. Consider the following four situations: 1 The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A = πr^2. With each positive number r there is associated one value of A, and we say that A is a function of r.
Four Ways to Represent a Function
3 The cost C of mailing a first-class letter depends on the weight w of the letter. Although there is no simple formula that connects w and C, the post office has a rule for determining C when w is known.
Four Ways to Represent a Function 4 The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. Figure shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t the graph provides a corresponding value of a.
C. The cost of mailing a first-class letter depends on the weight of the letter. Although there is no simple formula that connects and , the post office has a rule for determining when is known. D. The vertical acceleration of the ground as measured by a seismograph during an earthquake is a function of the elapsed time Figure 1 shows a graph gener- ated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of the graph provides a corresponding value of.
Each of these examples describes a rule whereby, given a number ( , , , or ), another number ( , , , or ) is assigned. In each case we say that the second num- ber is a function of the first number.
AP C a
rtw t
IGURE 1 tion during rthquake
{cm/s@}
(seconds)
Calif. Dept. of Mines and Geology
5
50
10 15 20 25
a
t
100
30 _
t , a
t.
a
C w
w C
C w
Figure 1: Vertical ground acceleration during the Northridge earthquake
Function
A function f is a rule that assigns to each element in a set A exactly one element, called f (x), in a set B.
We usually consider functions for which the sets A and B are sets of real numbers. The set A is called the domain of the function.
The number f (x) is called the value of f at x.
The range of f is the set of all possible values of f as x varies throughout the domain.
Function
A symbol that represents an arbitrary number in the domain of a function f is called an independent variable.
A symbol that represents a number in the range of is called a dependent variable.
Function
The graph of f also allows us to picture the domain of f on the x−axis and its range on the y−axis as in Figure.
-coordinate of any point on the graph is , we from the graph as being the height of the graph above the e graph of also allows us to picture the domain of on the
e -axis as in Figure 5.
ƒ
x
0 x
y ƒ( x )
domain
range
y
y
f f
x , y y f x
Function
EXAMPLE 1 The graph of a function is shown in Figure 6. (a) Find the values of and_._ (b) What are the domain and range of?
SOLUTION (a) We see from Figure 6 that the point lies on the graph of , so the value of at 1 is. (In other words, the point on the graph that lies above x 1 is three units above the x -axis.) When x 5, the graph lies about 0.7 unit below the x -axis, so we estimate that . (b) We see that f x is defined when 0 x 7 , so the domain of f is the closed
f 5 0.
f f 1 3
1, 3 f
x
y
0
1 1
f
f 1 f 5
f
The graph of a function f is shown in Figure.
a) Find the values of f (1) and f (5). b) What are the domain and range of f?
