Functions

HG1M11-ENGINEERING MATHEMATICS 1

1. Definition and Properties of Functions

A function is a rule that assigns exactly one element in a set to each element

in a set . In this case, we write .

We call the set the domain of and the set of all values in is the range of . We

refer to as the independent variable and as the dependent variable.

Functions can be defined by simple formulas, such as

but in general, any correspondence meeting the requirement of matching exactly one

to each defines a function.

2. Polynomial Functions

A polynomial is any function that can be written in the form

where are real numbers, called the coefficients of the polynomial,

with and is an integer (the degree of the polynomial).

Some simple polynomials

, a polynomial of degree 0 or constant,

, a polynomial of degree 1 or linear polynomial,

, a polynomial of degree 2 or quadratic polynomial,

, a polynomial of degree 3 or cubic polynomial,

and quartic and quintic polynomials are of degree 4 and 5 respectively.

3. Rational functions

Any function that can be written in the form

where and are polynomials with for any , is called a rational

function.

A simple rational function:

Note that the domain of this function is .

4. Trigonometric Functions

The trigonometric functions (also called circular functions) are functions of an angle.

They are used to relate the angles of a triangle to the lengths of the sides of a triangle.

Trigonometric functions are important in the study of triangles and modeling periodic

phenomena, among many other applications.

The most familiar trigonometric functions are the sine, cosine, and tangent. In the

context of the standard unit circle with radius 1, where a triangle is formed by a ray

originating at the origin and making some angle with the x-axis, the sine of the angle

gives the length of the y-component (rise) of the triangle, the cosine gives the length of

the x-component (run), and the tangent function gives the slope (y-component divided

by the x-component).

Fig. 1 - Sine and cosine of an angle θ defined using the unit circle.

Trigonometric functions are commonly defined as ratios of two sides of a right triangle

containing the angle, and can equivalently be defined as the lengths of various line

segments from a unit circle.

In this right triangle:

and .

It is noted that

for all .

Reciprocal of trigonometric functions:

For angles greater than 2π or less than −2π, simply continue to rotate around the circle; sine

and cosine are periodic functions with period 2π:

while tan is periodic with period :

,

for any angle θ and any integer k.

The smallest positive period of a periodic function is called the primitive (fundamental)

period of the function. The primitive period of the sine or cosine is a full circle, i.e. 2π

radians or 360 degrees.

The sine and cosine functions graphed on the Cartesian plane.

Notice that you could slide the graph of slightly to the left and get an exact copy of

the graph of . Specifically, we have the relationship

tan x = sin x/cos x cot x = cos x/sin x

sec x = 1/cos x csc x = 1/sin x

List several important identities.

For any real number and , the following identities hold:

5. Function composition

A composite function can be visualized as the combination of two "machines". The first takes input and outputs . The second takes and outputs .

The function composition of two or more functions takes the output of one or more functions as the input of others The functions ƒ X Y and g: Y Z can be composed by first applying ƒ to an argument x to obtain y ƒ x) and then applying g to y to obtain z = g(y). The composite function formed in this way from general ƒ and g may be written

This notation follows the form such that

Example 1

Given and , then

Find .

Example 2

For

and , show that

5. Inverse Functions

Let ƒ be a function whose domain is the set X, and whose range is the set Y Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:

If ƒ is invertible the function g is unique; in other words, there can be at most one function g satisfying this property. That function g is then called the inverse of ƒ denoted by ƒ 1 ( i.e. ).

Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, in which case the inverse relation is the inverse function.

Alternatively, the inverse if the function (if exists), can be determine by the inverse identity relation:

Not all functions have an inverse. For this rule to be applicable, each element y Y must correspond to no more than one x X a function ƒ with this property is called one-to-one, or information-preserving, or an injection

Example 3

All linear functions are invertible. Indeed,

In another word, the inverse of is defined by (replace by , since and are dummy

variables)

Alternatively, using the identity relation,

Quadratic functions are not invertible.

But the function violating the definition of a function. Why? Therefore, inverse does

not exists.

Class Work

Determine whether or not the following functions are invertible. Give reason(s). If

invertible, write their inverses.

(i) , for all .

(ii)

.

(iii) .

Inverse of trigonometric functions

We denote the inverse of by , respectively.

Thus. for

, we have

,

,

6. Limit of Functions

Limit at a any point

A limit of a function at is written as :

We want to find the limit of as x approaches a. To do this, we try to make the values of close to the limit L, by taking x values that are close to, but not equal to, a. In short, approaches L as x approaches a.

Example 4

Find (i)

, (ii)

Solution: (i)

This means as , the function

(ii) The direct substitution of yields a 0 to the denominator which make the

function undefined. In this case, we factorize and simplify the given expression.

Thus,

Exercise

Limits at infinity

A limit of a function at infinity is written as :

Means as , the function .

Example 5

Find (i)

, (ii)

Solution:

(i)

.

(ii)

(indeterminate). So, in this case, we divide both numerator

and denominator with (the highest order) to get

Hence,

The Squeeze Theorem

The squeeze theorem is an important concept that will be very helpful in upper year calculus

courses. The squeeze theorem states:

In simpler terms, the squeeze theorem states that if the graph of g

is squeezed between the graphs of f and h when x is near a, and if f

and h have the same limit L as x approaches a, then the limit of g

as x approaches a is also L. The graph to the right illustrates the

squeeze theorem.

Example 6

Solution: We know that

Therefore

Clearly, .

Hence, by the Squeeze Theorem, we have

6. Curve Sketching (http://www.nipissingu.ca/calculus/tutorials/curves.html)

Curve sketching is another practical application of differential calculus. We can make a fairly accurate sketch of any function using the following concepts.

Increasing and Decreasing Functions

The derivative of a function can tell us where the function is increasing and where it is decreasing. If

a) on an interval I, the function is increasing on I. b) on an interval I, the function is decreasing on I.

The intervals of increase and decrease will occur between points where f'(x) = 0 or f'(x) is undefined. However, these points are not necessarily critical numbers because we include x even if it is not in the domain of f. We simply want to find the intervals of increase and decrease around x, even if the function is not defined at that point.

The graph to the right illustrates this theorem.

From A to B, the slope of the tangent lines are all

negative, so the derivative, f'(x) is negative from A

to B. The theorem above states that the function is

decreasing from A to B. The graph shows that the

values of the function are decreasing between A

and B. Similarly, the function is also decreasing

between C and D. From B to C however, the slopes

of the tangent lines are positive. Therefore, the derivative is positive from B to C. The

graph shows that the values of the function are increasing between B and C.

The First Derivative Test

Let c be a critical number of a continuous function f. If

a) changes from positive to negative at c, there is a local maximum at c. b) changes from negative to positive at c, there is a local minimum at c. c) f' does not change sign at c, (that is, the derivative is positive before and after c or negative before and after c) there is no maximum or minimum at c.

The graphs below illustrate the first derivative test.

Concavity and Points of Inflection

A graph is called concave upward (CU) on an interval I, if the graph of the function lies above all of the tangent lines on I. A graph is called concave downward (CD) on an interval I, if the graph of the function lies below all of the tangent lines on I.

The second derivative of a function can tell us whether a function is concave upward or concave downward. If

a) for all x in an interval I, the graph is concave upward on I. b) for all x in an interval I, the graph is concave downward on I.

The intervals of concavity will occur between points where or is undefined. We test the concavity around these points even if they are not included in the domain of f.

The graphs below illustrate the different forms of concavity. Remember that there are two ways in which a graph can be concave upward or concave downward. Graphs A and C illustrate the types of concavity when the function is increasing on the interval, while graphs B and D illustrate concavity when the function is decreasing on the interval. These 4 graphs cover every different form of concavity.

A point P on a curve is called a point of inflection if the function is continuous at that point and either

a) the function changes from CU to CD at P b) the function changes from CD to CU at P

Points of inflection may occur at points where f''(x) = 0 or f''(x) is undefined, where x is in the domain of f. We must test the concavity around these points to determine whether they are points of inflection.

The graph to the right illustrates a curve with a point of inflection.

The Second Derivative Test

Let be a continuous function near c. If

a) and , then has a local minmum at c. b) and , then f has a local maximum at c.

The graphs containing local maximums and minimums in the "Increasing and Decreasing Functions" and "The First Derivative Test" sections above illustrate the second derivative test. When a graph has a local minimum, the function is concave upward (and thus lies above the tangent lines) at the minimum. Similarly, the function is concave downward at a local maximum.

Example 4

Example 5

Asymptotes

Before continuing with asymptotes, it is recommended that you review the vertical asymptote and infinite limits section of the limits tutorial at the link below.

Vertical Asymptotes and Infinite Limits

In order to properly sketch a curve, we need to determine how the curve behaves as x approaches positive and negative infinity. We must find the limit of the function as x approaches infinity.

For a function f defined on means that the values of approach the value L when x is taken to be sufficiently large.

For a function f defined on means that the values of approach the value L when x is taken to be sufficiently large, negatively.

The line y = L is called a horizontal asymptote of if either,

In order to find the horizontal asymptotes of a function, we use the following theorem. If n is a positive number, then

If n is a positive, rational number such that xn is defined for all x, then

Functions do not always approach a value as x approaches positive or negative infinity. Often there is no horizontal asymptote and the functions have infinite limits at infinity.

For example, the function approaches infinity when x is taken to be sufficiently large, positively or negatively.

Slant Asymptotes

Some curves may have an asymptote that is neither vertical nor horizontal. These curves approach a line as x approaches positive or negative infinity. This line is called the slant asymptote of the function. The graph to the right illustrates the concept of slant asymptotes.

If

then the function has a slant asymptote of y = mx + b.

Rational functions will have a slant asymptote when the degree of the numerator is one more than the degree of the denominator. To find the equation of the slant asymptote, we divide the numerator by the denominator using long division. The quotient will be the equation of the slant asymptote. The remainder is the quantity f(x) - (mx + b). We must show that the remainder approaches 0, as x approaches positive or negative infinity. The example below will give you a better idea of how to find the slant asymptote of a function.

Example 6

Evaluate lim ln .

Example 6

(Ans: 3)

Example

Example 7

(Ans: )

Example 8

(Ans: )

Curve sketching problem

Here is a more challenging question without the solution:

Math Humour! Interesting theorems

Theorem: Every positive integer is interesting. Proof: By contradiction, assume that there exists an uninteresting positive integer. Then there must be a smallest uninteresting positive integer. But that's pretty interesting! Therefore a contradiction!

Salary Theorem: The less you know, the more you make.

Proof:

Fact #1: Knowledge is Power

Fact #2: Time is Money

We know that: Power = Work / Time

And since Knowledge = Power and Time = Money

It is therefore true that Knowledge = Work / Money

Solving for Money, we get:

Money = Work / Knowledge

Thus, as Knowledge approaches zero, Money approaches infinity, regardless of the

amount of Work done.

Pick up lines:

1. Honey, you're sweeter than 3.14

2. I'm not trying to be obtuse, but you're acute girl.

3. You and I would add up better than a Riemann sum.

4. I'll love you from here to infinity

5. You fascinate me more then the Fundamental Theorem of Calculus.

6. Are you a differential function? Because I'd like to be tangent to your curves!

7. I am equivalent to the Empty Set when you aren't with me.

An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn't care.