Chap 0

Representation and Types of Functions Chapter 0 August 1999

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In this section

In this section, you will review some concepts from algebra. We will define function. Each function has both a domain and a range. Functions can be described via a rule (usually an equation), numerically (through a list or table of values) or graphically. Besides knowing the basic concepts, it is essential that we become comfortable working with several common classes of functions. These include polynomial, trigonometric, exponential and logarithmic functions. Each of these is described. There are also a variety of ways that two functions can be combined to obtain new functions. In fact, the ability to recognize complex functions as combinations of simple functions is a skill that we hope to develop throughout the course.

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Introduction

The study of relationships between different quantities is central to modern science. In fact, we encounter such relationships in our everyday experiences. What is the cost of gasoline per gallon, or bananas per pound? How much cotton (corn, wheat) can we produce per acre cultivated? What is the relationship between the freezing (or boiling) point of water and the salinity/mineral content? The study of calculus is primarily the study of these relationships; in particular, we will study how the change in the amount (value) of one quantity affects the value of another (related) quantity. In most applications, many quantities may be related. (The cost of gasoline or bananas, for example, is affected by many variables.) In this course, we will usually limit our discussion to the simpler case of only 2 variables, one depending on the other. The ideas we develop here extend to the more general case, however, and you will see this when you take a course in multivariate or vector calculus.

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Definitions and examples

A function can be thought of as an input/output rule. It is a machine which accepts appropriate input and returns specific output. Definition 1: A function f is a rule which, to each x in a set A, assigns a unique f (x) in a set B. The set A is called the domain of the function. The set B is called the range of the function. Note that the function’s name is f (though we can use any convenient name), while f (x) represents the output the function assigns to the input x. We say that f (x) is the image of x under f . In mathematics we often study functions whose domain and range are sets of real numbers and which can be represented by equations. For example, the square function accepts any real number as input, and returns it’s square as output. We may say that x2 is the image of x under f , or more concisely, f (x) = x2 . In this example, x is simply a symbol which can be replaced by any element of the domain of the function (in this case, any real number). So,

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f (2) = 4, f (10) = 100, f (−3) = 9. This suggests that we might make a table listing input and output values. x 1 2 3 4 5

f (x) 1 4 9 16 25

This leads us to an alternate definition of a function. Definition 2: A function f is a set of ordered pairs (x, f (x)), whose first coordinates come from a set A (the domain of f ) and whose second coordinates come from a set B (the range of f ), with the property that, for each x ∈ A, there is only one f (x) ∈ B for which (x, f (x)) is in f . Questions (true or false): The range of the square function is the set {0, 1, 4, 9, ...} The set of ordered pairs {(1, 1), (2, −3), (−4, 1)} is a function. The daily high temperature is a function of the date. (For answers to these questions, see the end of this document.)

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Representation of functions

We represent functions in 3 primary ways: by equations, tables and graphs. We have already considered both equations and tables for the function defined by the equation f (x) = x2 , the “square” function. We may also present this function through a graph.

Figure 1: Graph of f (x) = x2 on the interval [−3, 3]. Note that we have only shown a small part of the graph of this function. We can specify what portion of the graph we wish to view. Here is another piece of the picture. Our choice of presentation, in this case, depends on what aspects of the function we wish to emphasize. In other cases, the presentation may depend on how we have “discovered” the function.

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More Examples and Applications

All of us have seen examples of functions in our mathematics courses. Some of the important classes will be reviewed here 2

Figure 2: Graph of f (x) = x2 on the interval [20, 25].

5.1

Polynomials and rational functions

A polynomial is a function f with f (x) = an xn + an−1 xn−1 + ... + a1 x + a0 , where aj is a constant, j = 0, 1, ..., n. The highest power that appears in the equation defining a polynomial is its degree. For example, f (x) = x3 + x2 − 1 is a third degree polynomial. The coefficient of the highest power is the leading coefficient of the polynomial (in this case, 1). A related class of functions are the rational functions, which can be constructed by taking roots and/or quotients of polynomials. A simple, but important, example is the square root function, defined by the rule √ f (x) = x. . Another important class of rational functions can be expressed as quotients of polynomials. For example, R(x) =

x2 + x − 3 . x3 + 1

More complicated examples can easily be constructed.

5.2

Trigonometric functions

The trigonometric functions are periodic; that is, they repeat after a certain length of “time”, called the period. An example is the sine function. Here is a plot of y = sin x on the interval [0, 2π]. Note what happens when we multiply the variable by a constant. Here is a graph of y = sin (2x) , also on the interval [0, 2π]. Note what happens when we multiply the entire function by a constant. Here is a plot of y = 2 sin x on [0, 2π]. The length of time required for a trigonometric function to complete a cycle is called its period. The maximum value (of a sine or cosine function) is its amplitude. Other trig functions you should be familiar with are the tangent, cotangent, secant and cosecant. 3

Figure 3: Graph of y = sin(x) on the interval [0, 2π].

Figure 4: Graph of y = sin (2x) on the interval [0, 2π].

5.3

New functions from old

Quite often, different functions may share many characteristics and be related through fairly simple algebraic relationships. We will illustrate with some simple examples. Perhaps the simplest example is parallel lines: However, other functions can be related in the same way. These graphs, too, are parallel. In general, given a function f (x), the construction of a new function g(x) by addition of a constant, g(x) = f (x) + c, produces a function with a graph shifted by the constant c in the vertical direction. We can produce functions shifted in the horizontal direction as well: Note that, in this case, the plot of the second function is shifted to the right two units. In general, the graph of a function g(x) = f (x + c) will be shifted by −c units in the horizontal direction. Each of these examples is a special case of composition of functions, a very important way of generating new functions from old. We turn to this idea next.

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Figure 5: y = 2 sin x on [0, 2π].

Figure 6: Parallel lines: equations differ by an additive constant.

5.4

Composition of functions and inverse functions

The composition f ◦ g of two functions f and g is a new function defined by the rule (f ◦ g) (x) = f (g(x)) . That is, we first apply the function g to x, then apply f to g(x). So, if f (x) = x2 and g(x) = sin x, we will have (f ◦ g) (x)

=

f (g(x))

=

f (sin x)

=

(sin x)

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Order is important: in this case, (g ◦ f ) (x)

= g (f (x))
= g x2
= sin x2 . 5

Figure 7: The graphs of y = x2 − 1 and y = x2 + 1.

Figure 8: The graphs of y = x2 and y = (x − 2)2 . In the previous section, we discussed both horizontal and vertical shifts of functions. Suppose that h(x) = x + c. Then the composite function (h ◦ f ) (x)

= h(f (x)) = f (x) + c

is the vertical shift of f, while the composite function (f ◦ h) (x)

= f (h(x)) = f (x + c)

is the horizontal shift of f. Two functions f and g are said to be inverses of each other if they undo each other: that is, if (f ◦ g) (x) = (g ◦ f ) (x) = x. For example, the function f (x) = x + 2 is the inverse of the function g(x) = x − 2. We note that not all functions have inverses. In order for a function to have an inverse, it must be one to one. This means that, for each y in the range of f, there is only one x in the domain of f for which f (x) = y. For example, 6

the square function f (x) = x2 does not have an inverse, since any positive number√has two distinct square roots. However, if we restrict the domain of f to the nonnegative reals, the function g(x) = x will be an inverse for f.

5.5

Exponential and logarithmic functions

An example of an exponential function is f (x) = 2x , not to be confused with x2 . Here is a plot of this function on the interval [−3, 3].

Figure 9: The graph of f (x) = 2x on the interval [−3, 3] The number 2 is called the base of this exponential function. We can consider exponential functions with any positive base. The most important is the natural exponential function exp, with base e = exp(1) ≈ 2.7182818284590452353602874713526624977572470937000 The inverse of the natural exponential function is called the natural logarithm, denoted ln(x), or by tradition, simply ln x. Since exp(x) > 0 for all x, the natural logarithm will have domain (0, ∞). Here is a plot of y = ln x on the interval [1/10, 10]. The graph of y = ln x has a vertical asymptote at x = 0 (the y− axis).

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Answers to true/false questions

1. The range of the square function is the set {0, 1, 4, 9, ...} False. While it is true that the numbers in this set represent the squares of the whole numbers, unless otherwise specified, the domain of any function is the set of all numbers for which the function (or formula for the function) makes sense. In this case, the square function has the formula x2 . We can plug any real number into this rule, and get out any numbergreater than or equal to 0. Therefore, the range of the square function is [0, ∞). 2. The set of ordered pairs {(1, 1), (2, −3), (−4, 1)} is a function. True. The answer follows from the fact that, to the first number in each ordered pair, their is only one second number. That is, no first number appears more than once. The domain of this function is the set {1, 2, −4} and the range is the set {1, −3} . Note that the number 1 is the image of both 1 and −4, but that’s OK. 3. The daily high temperature is a function of the date.

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Figure 10: y = ln x True. On any given day (in any one location), their is a single high temperature. Measuring that temperature precisely may not be easy, and it is certainly the case that temperatures can fluctuate, even across town. But, picking a single location to measure the temperature, it is the case that there is only high temperature on any day.

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