Functions

Functions

Students will determine if a given equation is a function using the vertical line test and evaluate functions given member(s) of the domain.

Compiled by : Motlalepula Mokhele

Student at the University of Johannesburg (2014)

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The Rule is ‘ADD 4’

Ahmed

Peter

Ali

Jaweria

Hamad

Paris

London

Dubai

New York

Cyprus

Has Visited

There are MANY arrows from each person and each place is related to MANYPeople. It is a MANY to MANY relation.

Bilal

Peter

Salma

Alaa

George

Aziz

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Person Has A Mass of Kg

In this case each person has only one mass, yet several people have the same Mass. This is a MANY to ONE relationship

Is the length of

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30

Pen

Pencil

Ruler

Needle

Stick

cm object

Here one amount is the length of many objects.This is a ONE to MANY relationship

FUNCTIONS

Many to One Relationship

One to One Relationship

Domain Co-domain

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123456789

Image Set (Range)

x2x+1A B

f : x x2 4

fxx2 4

The upper function is read as follows:-

‘Function f such that x is mapped onto x2+4

Lets look at some functionType questions

If fx x 2 4 and gx 1 x 2

Find f2

Find g3

fx x2 42 2 = 8 gx 1 x2

3 3

= -8

Consider the function fx 3x 1 We can consider this as two simpler functions illustrated as a flow diagram

Multiply by 3 Subtract 13x 3x 1x

Consider the function f : x 2x 52

xMultiply by 2 Add 5

2x 2x 5Square

2x 52

f : x 3x 2 and gx : x x2Consider 2 functions

fg is a composite function, where g is performed first and then f is performedon the result of g.The function fg may be found using a flow diagram

xsquare

x2

Multiply by 33x2

Add 23x2 2

g f

Thus fg = 3x2 2

g f

fg x

x2 3x 2

3x2 2

24 14

2

Consider the function fx 5x 23

Here is its flow diagram

xx5

x5 -2 fx 5x 23

Draw a new flow diagram in reverse!. Start from the right and go left…

Multiply by 5 Subtract 2 Divide by three

Multiply by threeAdd twoDivide by 5

x3 x3 x +23 x +25

f 1x 3x 25And so

(a)

(b)

(c) (d)

(a) and (c)

Functions and Their Graphs

Definition of Relation

Relation – a set of ordered pairs, which contains the pairs of abscissa and ordinate. The first number in each ordered pair is the x-value or the abscissa, and the second number in each ordered pair is the y-value, or the ordinate.

Domain is the set of all the abscissas, and

range is the set of all ordinates.

Relations

A relation may also be shown using a table of values or through the use of a mapping diagram.

Illustration:

Using a table. Using a mapping diagram.Domain

Range

0 11 22 33 44 57 8

Definition of Function

Function – a characteristic of set of values where each element of the domain has only one that corresponds with it in the range. It is denoted by any letter of the English alphabet.

The function notation f(x) means the value of function f using the independent number x.

Example 1a.

Given the ordered pairs below, determine if it is a mere relation or a function.

(0,1) , (1, 2), (2, 3), (3, 4), (4, 5), (7, 8)

Answer: For every given x-value there is a

corresponding unique y-value. Therefore, the relation is a function.

Example 1b.

Which relation represents a function?

A. {(1,3), (2, 4), (3,5), (5, 1)}

B. {(1, 0), (0,1), (1, -1)}

C. {(2, 3), (3, 2), (4, 5), (3, 7)}

D. {(0, 0), (0, 2)}

Answer: A

Example 1c.

Which mapping diagram does not represent a function?

A. B.

C. D.

Types of Functions

1.Increasing, Decreasing, and Constant Functions

Constantf (x1) < f (x2)

(x1, f (x1))

(x2, f (x2))

Increasingf (x1) < f (x2)

(x1, f (x1))

(x2, f (x2))

Decreasingf (x1) < f (x2)

(x1, f (x1))

(x2, f (x2))

A function is increasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) < f (x2).

A function is decreasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) > f (x2).

A function is constant on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) = f (x2).

Solutiona. The function is decreasing on the interval (-∞, 0), increasing on the

interval (0, 2), and decreasing on the interval (2, ∞).

Describe the increasing, decreasing, or constant behavior of each function whose graph is shown.

-5 -4 -3 -2 -1 1 2 3 4 5

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-1-2

-3

-4-5

-5 -4 -3 -2 -1 1 2 3 4 5

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-1-2

-3

-4-5

a.

b.

Example 8a.

Solution: b.• Although the function's equations are not given, the graph

indicates that the function is defined in two pieces. • The part of the graph to the left of the y-axis shows that the

function is constant on the interval (-∞, 0). • The part to the right of the y-axis shows that the function is

increasing on the interval [0,∞).

Describe the increasing, decreasing, or constant behavior of each function whose graph is shown.

-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

1

-1-2

-3

-4-5

-5 -4 -3 -2 -1 1 2 3 4 5

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3

2

1

-1-2

-3

-4-5

a.

b.

Example 8a.

Example 8b.

Describe the increasing,

decreasing, or constant behavior of each function whose

graph is shown.

Decreasing on (-∞, 0);Increasing on (0, ∞)

Example 8b.



graph is shown.

Increasing on (-∞, 2);Constant on (2, ∞)

Example 8c.



graph is shown.

Increasing on (-∞,∞)

2.Continuous and Discontinuous Functions

A continuous function is represented by a graph which may be drawn using a continuous line or curve, while a discontinuous function is represented by a

graph which has some gaps, holes or breaks (discontinuities).

3.Periodic Functions

A periodic function is a function whose values repeat in periods or regular intervals.

y = tan(x) y = cos(x)

A linear function is a function of the form f(x) = mx +b where m and b are real numbers and m ≠ 0.

Domain: the set of real numbersRange: the set of real numbers

Graph: straight lineExample: f(x) = 2 - x

5. Quadratic Functions

A quadratic function is a function of the form f(x) = ax2 +bx +c where a, b and c are real numbers and a ≠ 0.

Domain: the set of real numbersGraph: parabola

Examples: parabolas parabolas opening upward opening downward

The graph of any quadratic function is called a parabola. Parabolas are shaped like cups, as shown in the graph

below.

If the coefficient of x2 is positive, the parabola opens upward; otherwise, the parabola opens downward.

The vertex (or turning point) is the minimum or maximum point.

Graphs of Quadratic Functions

Evaluation of Functions

Example 2.

If f (x) = x2 + 3x + 5, evaluate:

a. f (2) b. f (x + 3) c. f (-x)

Solution a. We find f (2) by substituting 2 for x in the

equation.f (2) = 22 + 3 • 2 + 5 = 4 + 6 + 5 = 15

Thus, f (2) = 15.

Example 2.

Solutionb. We find f (x + 3) by substituting (x + 3) for x in

the equation.f (x + 3) = (x + 3)2 + 3(x + 3) + 5

If f (x) = x2 + 3x + 5, evaluate: b. f (x + 3)

Equivalently,

f (x + 3) = (x + 3)2 + 3(x + 3) + 5 = x2 + 6x + 9 + 3x + 9 + 5

= x2 + 9x + 23.

Example 2.

Solutionc. We find f (-x) by substituting (-x) for x in the

equation.f (-x) = (-x)2 + 3(-x) + 5

If f (x) = x2 + 3x + 5, evaluate: c. f (-x)

Equivalently,

f (-x) = (-x)2 + 3(-x) + 5 = x2 –3x + 5.

Example 3a.

Which is the range of the relation described by y = 3x – 8 if its domain is {-1, 0, 1}?

A) {-11, 8, 5} B) {-5, 0 5} C) {-11, -8, -5} D){0, 3, 5}

Operations on Functions

Sum, Difference, Product, and Quotient of Functions

Let f and g be two functions. The sum, the difference, the product , and the quotient are functions whose domains are the set of all real numbers common to the domains of f and g, defined as follows:

Sum: (f + g)(x) = f (x)+g(x)

Difference: (f – g)(x) = f (x) – g(x)

Product: (f • g)(x) = f (x) • g(x)

Quotient: (f / g)(x) = f (x)/g(x), g(x) ≠ 0

Example 4a.

Let f(x) = 2x+1 and g(x) = x2 - 2. Find a. (f + g) (x) c.(g – f) (x) e. (f / g) (x)b. (f – g) (x) d. (f ∙ g) (x) f. (g/f) (x)Solution:a. (f + g) (x) = f(x) + g( x) = (2x+1 )+ (x2 – 2) = x2 +

2x - 1b. (f – g)(x) = f(x) - g(x) = (2x+1) - (x2 - 2) = -x2 + 2x +

3c. (g – f)(x) = g(x) - f(x) = (x2 - 2) – (2x +1) = x2 - 2x - 3d. (f ∙ g)(x) = f(x) ∙ g(x) = (2x+1)(x2 - 2) = 2x3 + x2 -

4x - 2e. (f/g)(x) = f(x)/g(x) = (2x+1)/(x2 - 2), f. (g/f)(x) = g(x)/f(x) = (x2 - 2)/(2x +1),

2x

2

1x

Example 5a.

Given f (x) = 3x – 4 and g(x) = x2 + 6,

find: a. (f ○ g)(x) b. (g ○ f)(x) Solution

a. We begin with (f o g)(x), the composition of f with g. Because (f o g)(x) means f (g(x)), we must replace each occurrence of x in the

equation for f by g(x).

f (x) = 3x – 4(f ○ g)(x) = f (g(x)) = 3(g(x)) – 4

= 3(x2 + 6) – 4 = 3x2 + 18 – 4

= 3x2 + 14

Thus, (f ○ g)(x) = 3x2 + 14.

Solution b. Next, we find (g o f )(x), the composition of g with f.

Because (g o f )(x) means g(f (x)), we must replace each occurrence of x in the equation for g by f (x).

g(x) = x2 + 6(g ○ f )(x) = g(f (x)) = (f (x))2 + 6

= (3x – 4)2 + 6 = 9x2 – 24x + 16 + 6

= 9x2 – 24x + 22

Notice that (f ○ g)(x) is not the same as (g ○ f )(x).

Example 5a.

Given f (x) = 3x – 4 and g(x) = x2 + 6, find: a. (f ○ g)(x) b. (g ○ f)(x)

Graphs of Relations and Functions

Graph of a Function

If f is a function, then the graph of f is the set of all points (x,y) in the Cartesian plane for which (x,y) is an ordered pair in f.

The graph of a function can be intersected by a vertical line in at most one point.

Vertical Line Test

If a vertical line intersects a graph more than once, then the graph is not the graph of a function.

Example 6a.

Determine if the graph is a graph of a function or just a graph of a relation.

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-4

5 10 15

graph of a

relation

Example 6b.


graph of a

function

Example 6c.


graph of a

relation

Example 6d.

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15 10 5 5 10 15 20 25

A


graph of a

relation

Example 6e.


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-4

-6 -4 -2 2 4 6

graph of a

relation

Example 6f.

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-10 -5 5 10


graph of a

relation

Example 6g.


graph of a

function

3 1 -3 -2 -1 1 2 3 4 -1 -2 -3 -5

Graphing Parabolas Given f(x) = ax2 + bx +c

4. Find any x-intercepts by replacing f (x) with 0. Solve the resulting quadratic equation for x.

5. Find the y-intercept by replacing x with zero.

6. Plot the intercepts and vertex. Connect these points with a smooth curve that is shaped like a cup.

Graphing Parabolas Given f(x) = ax2 + bx +c

1. Determine whether the parabola opens upward or downward. If a 0, it opens upward. If a 0, it opens downward.

2. Determine the vertex of the parabola. The vertex is

The axis of symmetry is

The axis of symmetry divides the parabola into two equal parts such that one part is a mirror image of the other.

a

bac

a

b

4

4,

2

2

a

bx

2

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Functions

Education

Transcript of Functions