Engineering Mathematics 1: Functions, Summaries of Engineering Mathematics

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ENGINEERING FUNCTIONS

Mr GM Tshitenge

CPUT

February 21, 2024

Introduction

The study of functions is central to engineering mathematics. Functions can be used to describe the way quantities change: for example, the variation in the voltage across an electronic component with time, the variation in position of an electric motor with time and the variation in the strength of a signal with both position and time. When trying to understand a mathematical function it is always useful to sketch a graph in order to obtain an idea of its behaviour. Students are encouraged to sketch a graph in order to obtain an idea of its behaviour. Graphics calculators are now readily available and they make this task relatively easy. Software packages are also available to allow such plots to be carried out on a computer.

Numbers and intervals cont...

It is useful to introduce the factorial notation. We write 3! to represent the product 3 × 2 × 1. THe expression 3! is read as "factorial 3". Similarly 4! is a shorthand way of writing 4 × 3 × 2 × 1. In general, for any positive integer, n, we can write

n! = n(n − 1 )(n − 2 )(n − 3 )...( 3 )( 2 )( 1 )

It is useful to represent numbers by points on the real line. Numbers which can be represented by points on the real line are known as real numbers. The set of real numbers is denoted by R. The set comprises all the rational and all the irrational numbers. The real line extends indefinitely to the left and to the right so that any real number can be represented.

Numbers and intervals cont...

Sometimes we are interested in only a small section, or interval, of the real line. We write [ 1 , 3 ] to denote all the real numbers between 1 and 3 inclusive, that is 1 and 3 are included in the interval. Thus the interval [ 1 , 3 ] consists of all real numbers x, such that 1 ≤ x ≤ 3.The square brackets, [ ], are used to denote that the end-points are included in the interval and such an interval is said to be closed. The interval ] 1 , 3 [ consists of all real numbers x, such that 1 < x < 3. In this case the end-points are not included in the interval and the interval is said to be open. Brackets, ] [, denote open intervals. An interval may be open at one end and closed at the other. For example ] 1 , 3 ] is open at the left and closed at the right. It consists of all real numbers x, such that 1 < x ≤ 3, and is known as a semi-open interval. Open and closed intervals can be represented on the real line. A closed end-point is denoted by • ; an open end-point is denoted by ◦.

Basic concepts of functions

A variable is a symbol selected to represent any one of a given set of numbers, here assumed to be real numbers. Should the set consist of just one number, the symbol representing it is called a constant. The range of a variable consists of the totality of numbers of the set which it represents. For example, if x is a day in September, the range of x is the set of positive integers { 1 , 2 , 3 , ..., 30 }; if x (m-meter) is the length of rope cut from a piece 50m long, the range of x is the set of numbers greater than 0 and less than 50. A correspondence (x, y ) between two sets of numbers which pairs to an arbitrary number x of the first set exactly one number y of the second set is called a function. In this case, it is customary to speak of y as a function x. The variable x is called the independent variable and y is called the dependent variable.

Basic concepts of functions cont...

A function may defined By a table of correspondents or table of values

Table 1: Table of values x 1 2 3 4 5 6 7 8 9 10 y 3 4 5 6 7 8 9 10 11 12

By an equation or formula, as y = x + 2. For each value assigned to x, the above relation yields a corresponding value for y. Note that the table above is a table of values for this function. A function is called single-valued if , to each value of y in its range, there corresponds just one value x; otherwise, the function is called multivalued. For example, y = x + 3 defines y as a single-valued function of x while y = x^2 defines y as a multivalued (here, two values) function of x.

One-to-many

Some rules relating input to output are not functions. Consider the rule: "take plus or minus the square root of the input", that is

x → ±

x

Now, for example if 4 is the input, the output ±

4 which can be 2 or −2. Thus a single input has produced more than one output. The rule is said to be one-to-many, meaning that one input has produced many outputs. Rules with this property are not functions. For a rule to be a function there must be a single output for any given input. By defining a rule more specifically, it may become a function. For example, consider the rule: "take the positive square root of the input". This rule is a function because there is a single output for a given input. Note that the domain of this function is [ 0 , ∞[ and the range is also [ 0 , ∞[.

Many-to-one and one-to-one functions

If the function y = f (x) is such that for every y in the range there is one and only one x in the domain such that y = f (x), we say that f is a one-to-one correspondence. Functions that are one-to-one correspondences are sometimes called bijections. Note that all functions of the form ax + by + c = 0 are bijections. Note that y = x^2 is not a bijection. The inputs 2 and −2 both produce the same output, 4, and the function is said to be many-to-one. This means that many inputs produce the same output.

Many-to-one and one-to-one functions cont...

Figure 2: h(x) one-to-one function

A function is one-to-one if different inputs always produce different outputs. A horizontal line will intersect the graph of a one-to-one function in only one place. Figure 2 illustrates a one-to-one function, h(x).

Parametric definition of a function

Functions are often expressed in the form y (x). For every value x the corresponding value of y can be found and the point with coordinates (x, y ) can then be plotted. Sometimes it is useful to express x and y coordinates in terms of a third variable known as a parameter. Commonly we use t or θ to denote a parameter.Thus the coordinates (x, y ) of the points on a curve can be expressed in the form

x = f (t) y = g (t)

For example, given the parametric equations

x = t^2 y = 2 t 0 ≤ t ≤ 5

we can calculate x and y for various values of the parameter t.

Composition of functions

Consider the function y (x) = 2 x^2. We can think of y (x) as being composed of two functions. One function is described by the rule: "square the input", while the other function is described by the rule: "double the input". This shown in Figure 4.

Figure 4: The function: y (x) = h(g (x))

Mathematically, if h(x) = 2 x and g (x) = x^2 then y (x) = 2 x^2 = 2 (g (x)) = h(g (x))

Composition of functions cont...

The form h(g (x)) is known as a composition of functions h and g , it is also denoted: (h ◦ g )(x) = h(g (x)) (1) Note that the composition h(g (x)) is different from g (h(x)). For example, if f (t) = 2 t + 3 and g (t) = t+ 2 1 ,then

(f ◦ g )(t) = f (g (t)) = f

t + 1 2

= t + 1 + 3 = t + 4 (g ◦ f )(t) = g (f (t) = g ( 2 t + 3 )

= 2 t + 3 + 1 2

= t + 2

Clearly (f ◦ g )(t) 6 = (g ◦ f )(t)

Inverse of a function cont...

Given g (x) = x− 2 1 find the inverse of g.

y = x − 1 2 2 y = x − 1 x = 2 y + 1 ⇐⇒ g −^1 (x) = 2 x + 1

Mr GM Tshitenge (CPUT) Figure 6:DEECE EMA156S/157S^ Inverse function graph February 21, 2024 19 / 66

Exponential functions: Terminology

An exponent is another name for a power or index. Expression involving exponents are called exponential expressions, for example 3^4 , ab, mn. In the exponential expression ax^ , a is called the base; x is the exponent. Exponential expressions can be simplified and manipulated using the laws of indices. These laws are summarised here. Laws of indices

aman^ = am+n^ (4) am an^ = am−n^ (5) a^0 = 1 (6)

a−m^ =

am^

(am)n^ = (an)m^ = amn^ (8)