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1

INTRODUCTION

A Short Review About Functions of Real Variable
Stupid Stuff I Wish Someone Had Told Me Four Years Ago (Prepared by Roosevelt Moreno PhD - 2012)

1

Introduction

The fundamental objects that we deal with in calculus are functions. This short document prepares the way for calculus by discussing the basic ideas concerning functions, their graphs, and ways oftransforming and combining them. We stress that a function can be represented in different ways: by an equation, in a table, by a graph, or in words. We look at the main types of functions that occur in calculus and describe the process of using these functions as mathematical models of real-world phenomena. Lets review some of the concept we are learning until now: A function f is a rule that assigns to eachelement x in a set D exactly one element, called f(x), in a set E. In our context the sets D and E are sets of real numbers. The set D is called the domain of the function. The number f(x) is the value of f at x and is read “f of x ”. The set E is called the range of the function. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. Asymbol that represents a number in the range of a function f is called a dependent variable. The most common method for visualizing a function is its graph. If f is a function with domain D, then its graph is the set of ordered pairs: {(x, f (x)|x)} (Notice that these are input-output pairs.) In other words, the graph of f consists of all points (x, y) in the coordinate plane such that y = f (x) and xis in the domain of f . The graph of a function f gives us a useful picture of the behavior of f. Since the y-coordinate of any point (x, y) on the graph is y = f (x), we can read the value of f (x) from the graph as being the height of the graph above the point x (see Figure 1a). 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 1b.y

( x, f (x))
range

y

f (x)
f (1)
1 2

y = f (x)

f (!2)
2 1

x

x

0

x

domain
(b) Domain and Range visualization

(a) Heights above x-point

Figure 1: Function representation

1

2

TRANSFORMATIONS AND SYMMETRY OF GRAPHS

2
2.1

Transformations and Symmetry of Graphs
Rigid Transformations

Rigid transformations do not change the shape of the graphof a function. 2.1.1 Reflection

A reflection is a mirror image. Placing the edge of a mirror on the x-axis will form a reflection in the x-axis. This can also be thought of as “ folding” over the x-axis. If the original (parent) function is y = f (x), the reflection over the x-axis is function −f (x), as we can see in Figure 2a. Placing the edge of a mirror on the y-axis will form a reflection inthe y-axis. This can also be thought of as “folding” over the y-axis. If the original (parent) function is y = f (x), the reflection over the y-axis is function f (−x) as shown in Figure 2b.
20

y

4

f (x)
(3, 9)
4 2 10

3

y
( 11 , 2) 2

(! 11 , 2) 2
2 4

2

x

(!3, !9)

g(x)
10

1

f (x)
2 4 6

g(x)

6

4

2 1

x

20

(a) Reflection over the x-axis(b) Reflection2 over the y-axis

Figure 2: Reflection of functions

2.1.2

Translations

If the original (parent) function is y = f (x), the translation (sliding) of the function horizontally to the left or right is given by the function f (x − h), as shown in Figure 3a. If h > 0, the graph translates (slides) to the right. If h < 0, the graph translates (slides) to the left. In any of thecases above, we are “subtracting” the value of h from x. Thus f (x+2) is really f (x−(−2)) and the graph moves to the left.
y
4
6

f (x) = x + 4

y
3

(0, 4)4
+4
f (x) = x
2

f (x) = x

h(x) = x + 4

2
2

(0, 0)
2

2

4

6

8

1

g(x) = x ! 5
+5

!5

x

!4

(!4, 0)
4 2

4

(0, 0)

2

4

(5, 0)

6

8

10

x

(0, !5)
6

f (x) = x ! 5...
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