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Topic 6: Functions
 

 

1. DEFINITIONS

A function is any relationship which takes one element of one set and assigns it to one and only one element of a second set. The first set is called the domain of the function and the second set is called the co-domain.

Mathematically we say that each element of the domain is mapped onto its image in the co-domain. The range is the set of all images. The range is the subset of the co-domain.

The above function forms the ordered pairs:

Example

The domain is , the co-domain is and the range is

A function f can associate with more than one element of the domain onto the same element of the range. Such functions are said to be many-to-one.

Functions for which each element of the domain is associated onto a different element of the range are said to be one-to-one.

Relationships which are one-to-many can occur, but from our preceding definition, they are not functions.

Example

One-to-one (a function)

Many to one (a function)

One to many or many to many (not a function)

The function notation

We shall write f(x) to represent the image of x under the function f. The letters commonly used for this purpose are f, g and h.

Example

Given that , find the values of

Solution

Notice:

can also written as which read as “f is a function which maps x onto

2. CLASSIFICATION OF FUNCTIONS

Constant function

A function that assigns the same value to every member of its domain is called a constant function. The constant function that assigns the value c to each real number is sometimes called the constant function c.

Example

The function given by is constant

Polynomial function

A polynomial function is expressible in the form where n is a nonegative integer and are real constants.

Example

The following functions are polynomial functions:

Rational function

A function that is expressed as a ratio of two polynomial is called a rational function and it has the form

Example

The following functions are rational functions:

Irrational function

A function that is expressed as root extractions is called irrational function and it has the form , where is a polynomial and n is a positive integer greater or equal to 2.

Example

The following functions are irrational functions

Explicit algebraic function

An explicit algebraic function is a function in which the dependent variable has been given explicitly in terms of the independent variable and can be evaluated using finitely many additions, subtractions, multiplications, divisions and root extractions.

Also, polynomials and rational functions are explicit algebraic functions. All remaining functions fall into 2 categories implicit algebraic functions and transcendental functions.

Example

The following functions define explicit algebraic functions

Implicit algebraic functions and transcendental functions

An implicit algebraic function is a function in which the dependent variable has not been given explicitly in terms of the independent variable. It is denoted by .

Transcendental function is a function that does not expressible as a finite combination of the algebraic operations of addition, subtraction, multiplication, division, raising to a power and extracting a root. Transcendental functions include the exponential functions, the logarithm functions and the trigonometric functions.

Example

The following functions are implicit functions

The following functions are transcendental functions

3. FINDING DOMAIN OF DEFINITION

Case 1: The given function is a polynomial

Given that is polynomial, then the domain of definition is the set of real numbers. That is

Case 2: The given function is a rational function

Given that where g(x) and h(x) are polynomials, then the domain of definition is the set of real numbers excluding all values where the denominator is zero. That is

Case 3: The given function is an irrational function

Given that where g(x) is a polynomial, there are two cases:

a) If n is odd number, then the domain is the set of real numbers. That is
b) If n is even number, then the domain of definition is the st of all values of x such that g(x) is positive or zero. That is

Example

Find the domainof the finition of the function

Solution

The domain of the function is the set of real numbers since the function is a polynomial.

Example

Find the domain of definition of the function

Solution

Condition:

Example

Find the domain of definition of the function

Solution

Condition:

We need to construct the sign table

Thus,

Example

Find domain of definition of the function

Solution

Since the index in radical sign is odd number, then

4. LINEAR FUNCTIONS

The function represents a straight line in xy plane. This function is called a linear function. Its graph is the set of points (x,y) for which . If m = 0 then for all x and f is called constant function whose graph is a horizontal line. m is called the gradient (or slope) of and b is the y-intercept.

The gradient m is the rate of change of the linear function f. If m is positive, the change in f(x) is an increase; if m is negative, the change f(x) in is a decrease. The rate of change of f is given by dividing the change in f(x) by the change in x. That is given two points and then

Notice:

• In the equation , if m = 0, then the equation becomes y = b. For any value of x, the corresponding value of y is b. By connecting the coordinates, we obtain a line parallel to the x-axis. Thus, the equation of horizontal line is of the form y = b (a relation containing y only).

• In the equation , by interchanging x and y and putting m = 0, we obtain x = b. For any value of y, the corresponding value of x is b. By connecting the coordinates, we obtain a line parallel to the y-axis. Thus, the equation of a vertical line is of the form x = b (a relation containing x only).

Example

Find formula for the linear function f if

Solution

Since f is linear, let

The gradient is

Now, but , then and then

Graph

x 2 6
y 1 13

Example

Find the equation of a straight line passing through the points (1,1) and (2,3)

Solution

Method 1

Let

But

Method 2

 

5. QUADRATIC FUNCTIONS

A quadratic function has the form and represents a parabola in Cartesian plane. We may write . A parabola has one axis of symmetry and a vertex. It has no centre. The vertex of the parabola is also called the turning point.

The vertex of the parabola is given by and the axis of symmetry is given by .

The concavity of the parabola depends on sign of a.
• If a is positive, the concavity is upwards and
• if a is negative, the concavity is downwards.

If the concavity is upwards the turning point is the minimum point of the parabola and if the concavity is downwards the turning point is the maximum point. To sketch a parabola in Cartesian plane, we need to find some additional points. If the interval to be used is not given we choose an interval such that the vertex will be included in that chosen interval.

Example

Consider the parabola . Find
a) Vertex
b) Axis of symmetry
c) Explain the concavity
d) Is the turning point the minimum or maximum point?
e) Sketch the curve.

Solution

Here

a) . Thus, vertex is (0,0)
b) Axis of symmetry is
c) The concavity is upwards because a is positive
d) Since the concavity is upwards then the turning point is the minimum point.
e) Curve: Since the vertex is (0,0), let take for example the interval [-3,3]

Additional point:

Curve

6. COMPOSITE FUNCTION

This combined or composite function of functions f and g is written as or simply . The function f is performed first and so is written nearer to the variable x.

Example

If and , find

a)               b)

Solution

a)

b) and then

Remark:

Clearly from the above example we see that

7. INVERSE FUNCTION

The inverse function of the function f, denoted , is the function that maps the range of function f onto its domain. Only one-to-one functions can have inverse functions. To find the inverse of one-to-one functions we will change the subject of a formula.

Example

Find the inverse of the function

Solution

Let . We need to find x

Then,

Example

Find the inverse of the function

Solution

Let

Then,

8. LOGARITHMIC FUNCTIONS

 

9. TRIGONOMETRIC FUNCTIONS

Cosine and sine

sin x and cos x are functions which are defined for all positive and negative values of x even for x = 0. Thus, the domain of sin x and cos x is the set of real numbers. The range of sin x and cos x is .

Tangent and Cotangent

tan x is not defined for . Thus the domain of tan x is , the range is the set of real numbers.

cot x is not defined for Thus the domain of cot x is , the range is the set of real numbers.

Secant and Cosecant

Similar to tangent, the domain of sec x is .

Similar to cotangent, the domain of csc x is .

The range of sec x and csc x is .

Inverse sine and inverse cosine

sin x and cos x have inverse functions called inverse sine and inverse cosine denoted by .

In older literature, are called arcsine of and arccosine of sin x and cos x. They are denoted by respectively.

Notice

The inverses of the trigonometric functions are not functions, they are relations. The reason why they are not functions is that for a given value of x , there are an infinite number of angles at which the trigonometric functions take on the value of x . Thus, the range of the inverses of the trigonometric functions must be restricted to make them functions. Without these restricted ranges, they are known as the inverse trigonometric relations.

Because sin x (restricted) and are inverse each other; also cos x (restricted) and are inverses each other, it follow that:

Inverse tangent

Inverse teangent is denoted by and

Because tan x (restricted) and inverse tangent are inverse each other, it follow that

Inverse secant

 

10. ODD AND EVEN FUNCTIONS

A function f(x) is said to be odd if the following conditions are satisfied

The graph of such a function looks the same when rotated through half a revolution about 0. This is called rotational symmetry.

Example

The function is odd function since and

A function f(x) is said to be even if the following conditions are satisfied

Example

The function is even function since and

11. PERIODIC FUNCTIONS

A function f is called periodic if there is a positive number P such that whenever x and lie in the domain of f. We call P a period of the function.

The smallest positive period is called the fundamental period (also primitive period, basic period, or prime period) of f. A function with period P will repeats on intervals of length P, and these intervals are referred to as periods.

Specifically, a function is periodic with period P if its graph is invariant under translation in the x-direction by a distance of P. The most important examples of periodic functions are the trigonometric functions.

Any function which is not periodic is called aperiodic.

Example

For the sine and cosine functions, the period is since

Example

For tangent and cotangent functions the period is since

Example

Find the period of

Solution

Let P be that period:

Combining periodic functions

We have seen that sine and cosine are both periodic and have the same period. When we add them up, subtract them, multiply them, etc we get functions that are also periodic.

To find the period of a sum or a product of two different periodic functions, we find the Lowest Common Multiple (LCM) of two periods.

Example

Find the peiod of the function

Solution

For

For

Example

Find the period of the funvtion

Solution

12. APPLICATIONS

The

 

 

 
 

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