math103 mathematics for business and economics - i chapter 2 – functions and graphs

MATH103

Mathematics for Business and Economics - I

Chapter 2 – Functions and Graphs

2.1 Functions2.1 FunctionsThe idea of a function is this: a correspondence between two sets D and R such that to each element of the first set, D, there corresponds one and only one element of the second set, R.The first set is called the domain, and the set of corresponding elements in the second set is called the range.

Notation: if y is a function of x, we write ◦ y = f(x)◦Other common symbols for functions include but are not

limited to g, h, F, G

Function Evaluation

• Consider our function

• What does f (-3) mean? Replace x with the value –3 and evaluate the expression

• The result is 7 . This means that the point (-3,7) is on the graph of the function.

2( ) 2f x x

2( 3) ( 3) 2f

EXAMPLE 1 Evaluating a Function

Solution

Let g be the function defined by the equationy = x2 – 6x + 8.Evaluate each function value.

a. g 3 b. g 2 c. g1

2

d. g a 2 e. g x h

a. g 3 32 6 3 8 1

b. g 2 2 2 6 2 8 24

EXAMPLE 1 Evaluating a Function

Solution continued

c. g1

2

1

2

2

61

2

8

21

4

d. g a 2 a 2 2 6 a 2 8

a2 4a 4 6a 12 8

a2 2a

e. g x h x h 2 6 x h 8

x2 2xh h2 6x 6h 8

AGREEMENT ON DOMAIN

If the domain of a function that is defined by an equation is not explicitly specified, then we take the domain of the function to be the largest set of real numbers that result in real numbers as outputs.

Domain of a Function• Consider

which is not a real number. • Question: for what values of x is the function defined?

( ) 3 2f x x

0

(0) ?

( ) 3( ) 20 2

f

f

• Answer:

2

3x

is defined only when the radicand (3x-2) is equal to or greater than zero. This implies that

( ) 3 2f x x

Therefore, the domain of our function is the set of real numbers that are greater than or equal to 2/3.

• Example1: Find the domain of the function

• Answer:

1( ) 4

2f x x

8 , [8, )x x

• Example : Find the domain of

• In this case, the function is defined for all values of x except where the denominator of the fraction is zero. This means all real numbers x except 5/3.

1( )

3 5f x

x

EXAMPLE 3 Finding the Domain of a Function

Find the domain of each function.

a. f x 1

1 x2

Solution

a. f is not defined when the denominator is 0.

b. g x x

c. h x 1

x 1d. P t 2t 1

1 x2 0

x 1Domain: {x|x ≠ –1 and x ≠ 1}

, 1 1,1 1,

EXAMPLE 3 Finding the Domain of a Function

Solution continued

The square root of a negative number is not a real number and is excluded from the domain.

b. g x x

c. h x 1

x 1

Domain: {x|x ≥ 0}, [0, ∞)

The square root of a negative number is not a real number and is excluded from the domain, so x – 1 ≥ 0. However, the denominator ≠ 0.

EXAMPLE 3 Finding the Domain of a Function

Solution continued

So x – 1 > 0 so x > 1.

d. P t 2t 1

Domain: {x|x > 1}, or (1, ∞)

Any real number substituted for t yields a unique real number.

Domain: {t|t is a real number}, or (–∞, ∞)

Example #2a (p.80)

• Find the domain of f(x) = x/(x2 –x – 2)– The domain would be the set of all real numbers except

those values of x which set the denominator equal to zero– These values are found by factoring

• (x2 –x – 2) = (x + 1)(x - 2)• x = -1, 2

– So the domain is the set of all real numbers , except x =-1, 2

Equality of Functions• Two functions, f and g are equal (f = g) if

– The domain of f is equal to the domain of g– For every x in the domain of f and g, the values of

the two functions are the same; that is f(x) = g(x)

Example #1 (p. 79-80)• Which of the following functions are equal

– f(x) = (x + 2)(x + 1)/(x – 1)– g(x) = x + 2– h(x) = x + 2

• Domains of g, h, the set of all real numbers and are equal,

• but the domain of f is the set of all real numbers except x = 1

1

2A polynomial function of degree n is a function of the form

where n is a nonnegative integer and the coefficients an, an–1, …, a2, a1, a0 are real numbers with a ≠ 0.

f x an xn an 1xn 1 ... a2 x2 a1x a0 ,

3

4

2.3 Combinations of Functions2.3 Combinations of Functions

f

g

x f x g x , g x 0.(iv) Quotient

(i) Sum f g x f x g x

(ii) Difference f g x f x g x

(iii) Product fg x f x g x

(v)

EXAMPLE 1 Combining Functions

f(x) = x2 , g(x) = 3x, find;i. f(x) + g(x) = x2 + 3x

ii. f(x).g(x) = 3x3

iii. f(x) – g(x) = x2 – 3x

iv. f(x)/g(x) = x2/3x = x/3

v. cf(x) = cx2

EXAMPLE 2 Combining Functions

EXAMPLE 3 Combining Functions

Let f x x2 6x 8, and g x x 2.

Find each of the following functions.

a. f g x b. f g x

c. fg x d. f

g

x Solution

a. f g x f x g x x2 6x 8 x 2 x2 5x 6

EXAMPLE 3 Combining Functions

Solution continued

f x x2 6x 8 and g x x 2

b. f g x f x g x x2 6x 8 x 2 x2 7x 10

c. fg x x2 6x 8 x 2 x3 2x2 6x2 12x 8x 16

x3 8x2 20x 16

EXAMPLE 3 Combining Functions

Solution continued

f x x2 6x 8 and g x x 2

d. f

g

x f x g x , g x 0

x2 6x 8

x 2, x 2 0

x 2 x 4

x 2, x 2

COMPOSITION OF FUNCTIONSIf f and g are two functions, the composition of function f with function g is written asf og and is defined by the equation

f og x f g x ,

where the domain of values x in the domain of g for which g(x) is in the domain of f.

consists of thosef og

COMPOSITION OF FUNCTIONS

EXAMPLE 1 Evaluating a Composite Function

LetFind each of the following.

f x x3 and g x x 1.

a. f og 1 b. go f 1 c. f o f 1 d. gog 1

Solution

a. f og 1 f g 1 f 2 23

8

EXAMPLE 1 Evaluating a Composite Function

Solution continued

b. go f 1 g f 1 g 1 11 2

f x x3 and g x x 1

c. f o f 1 f f 1 f 1 1 3 1

d. gog 1 g g 1 g 0 0 1 1

EXAMPLE 2 Finding Composite Functions

LetFind each composite function.

f x 2x 1 and g x x2 3.

a. f og x b. go f x c. f o f x Solution

a. f og x f g x f x2 3 2 x2 3 1

2x2 6 1

2x2 5

EXAMPLE 2 Finding Composite Functions

Solution continued

b. go f x g f x g 2x 1 2x 1 2 3 4x2 4x 2

c. f o f x f f x f 2x 1 2 2x 1 1 4x 3

f x 2x 1 and g x x2 3.

EXAMPLE 3 Finding the Domain of a Composite Function

Let f x x 1 and g x 1

x.

c. Find f og x and its domain.

d. Find go f x and its domain.

b. Find go f 1 .a. Find f og 1 .

Solution

a. f og 1 f g 1 f 1 11 0

EXAMPLE 3 Finding the Domain of a Composite Function

f x x 1 and g x 1

x

c. f og x f g x f

1

x

1

x1

d. go f x g f x g x 1 1

x 1

b. go f 1 g f 1 g 0 not defined

Solution continued

Domain is (–∞, 0) U (0, ∞).

Domain is (–∞, –1) U (–1, ∞).

EXAMPLE 4 Decomposing a Function

Show that each of theLet H x 1

2x2 1.

following provides a decomposition of H(x).

a. Express H x as f g x , where f x 1

x and g x 2x2 1.

b. Express H x as f g x , where f x 1

x and g x 2x2 1.

EXAMPLE 4 Decomposing a Function

Solutiona. f g x f 2x2 1

1

2x2 1

H x b. f g x f 2x2 1

1

2x2 1

H x

An ordered pair of real numbers is a pair of real numbers in which the order is specified, and is written by enclosing a pair of numbers in parentheses and separating them with a comma.

The ordered pair (a, b) has first component a and second component b. Two ordered pairs (x, y) and (a, b) are equal if and only if x = a and y = b.The sets of ordered pairs of real numbers are identified with points on a plane called the coordinate plane or the Cartesian plane.

DefinitionsWe begin with two coordinate lines, one horizontal (x-axis) and one vertical (y-axis), that intersect at their zero points. The point of intersection of the x-axis and y-axis is called the origin. The x-axis and y-axis are called coordinate axes, and the plane formed by them is sometimes called the xy-plane.

The axes divide the plane into four regions called quadrants, which are numbered as shown in the next slide. The points on the axes themselves do not belong to any of the quadrants.

DefinitionsThe figure shows how each ordered pair (a, b) of real numbers is associated with a unique point in the plane P, and each point in the plane is associated with a unique ordered pair of real numbers. The first component, a, is called the x-coordinate of P and the second component, b, is called the y-coordinate of P, since we have called our horizontal axis the x-axis and our vertical axis the y-axis.

DefinitionsThe x-coordinate indicates the point’s distance to the right of, left of, or on the y-axis. Similarly, the y-coordinate of a point indicates its distance above, below, or on the x-axis. The signs of the x- and y-coordinates are shown in the figure for each quadrant. We refer to the point corresponding to the ordered pair (a, b) as the graph of the ordered pair (a, b) in the coordinate system. The notation P(a, b) designates the point P in the coordinate plane whose x-coordinate is a and whose y-coordinate is b.

EXAMPLE 1 Graphing Points

Graph the following points in the xy-plane:A 3,1 , B 2, 4 , C 3, 4 , D 2, 3 , E 3,0

Solution

A 3,1 3 units right, 1 unit up

3 units left, 4 units downC 3, 4 2 units left, 4 units upB 2, 4

3 units left, 0 units up or downE 3,0 2 units right, 3 units downD 2, 3

Slide 2.1- 42Copyright © 2007 Pearson Education, Inc. Publishing as

Pearson Addison-Wesley

EXAMPLE 1 Graphing Points

Solution continued

DefinitionsThe points where a graph intersects (crosses or touches) the coordinate axes are of special interest in many problems. Since all points on the x-axis have a y-coordinate of 0, any point where a graph intersects the x-axis has the form (a, 0). The number a is called an x-intercept of the graph. Similarly, any point where a graph intersects the y-axis has the form (0, b), and the number b is called a y-intercept of the graph.

PROCEDURE FOR FINDING THE INTERCEPTS OF A GRAPH

Step1 To find the x-intercepts of an equation, set y = 0 in the equation and solve for x.

Step 2 To find the y-intercepts of an equation, set x = 0 in the equation and solve for y.

EXAMPLE 1 Finding Intercepts

EXAMPLE 2 Finding Intercepts

Find the x- and y-intercepts of the graph of the equation y = x2 – x – 2.

SolutionStep 1 To find the x-intercepts, set y = 0, solve for x.

The x-intercepts are –1 and 2.

0 x2 x 2

0 x 1 x 2 x 1 0 or x 2 0

x 1 or x 2

EXAMPLE 2 Finding Intercepts

Solution continued

Step 2 To find the y-intercepts, set x = 0, solve for y.

y 02 0 2

y 2

The y-intercept is –2.

The following steps can be used to draw the graph of a linear equation.

The graph of the linear equation(line)

Step1 ) Select at least 2 values for xStep2 ) Substitute them in the equation and find the corresponding values for yStep3 ) Plot the points on cartesian planeStep4 ) Draw a straight line through the points.

X -2 -1 0 1

y -1 1 3 5

1

-2

-1

3

1

5

-1

x-intercept y=0 0=2x+3

x= 2/3

(2/3 , 0) is x-intercepty- intercept

x=0 y=2.0+3y= 3

(0,3) y-interceptx-intercept

y-intercept

3

-2

Step1

Step2

Step3

Step4

x-intercept x-intercept

y-intercept

Vertex point

5-23/2

-49/4

Step1

Step2

Step3

Step4 4 : 1 0, concave upStep a

Step5

Ex : Sketch the Graph of 2 4 8y x x

2

:

1: 2, 4

4 4 16 4( 1)( 8) 162, 4

2 2( 1) 4 4( 1) 4

Solution

step Vp

b b ac

a a

2

2

2 : intercept(s), 0

4 8 0

4 4( 1)( 8) 16 32 16 0 intercept(s)

this graph never cuts x-axis

Step x y

x x

no x

3: intercept, 0

8 (0, 8)

Step y x

y

4 : 1 0, concave downStep a

y-intercept

Vertex point

-8

2

4