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 , denoted by f, is a rule that

assigns to each element x in a set X exactly one element f(x) in a set Y.

The set X is called the domain of the function and Y its codomain.

The set of assigned elements in Y is called the range of the function f.

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

Example 1a. Given the set of ordered pairs below,

determine if it is a mere relation or a function.

1. {(0,1) , (1, 2), (2, 3), (3, 4), (4, 5), (7, 8)}

2. {(0,0) , (1,1), (1, 1), (2, 4), (2, 4)} 3. {(2,0) , (1,3), (2, 5), (1, 4), (3, 5)}

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)}

Example 1c. Which mapping diagram does not represent a

function? A. B.

C. D.

Evaluation of Functions

Example 2.If f (x) = x2 + 3x + 5, evaluate:

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

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}

Example 3b. Which is the range of the relation

described by 3y = 2x2 – 36 if its domain is {3, 6, 9}?

A) {-6, 12, 42} B) {6, 12, 42} C) {0, 6, 12} D) {-6, 0, 12}

Example 3c. Find the domain and range of each function. 1. 2.

4)( xxf22)(

x

xf

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)

Example 4b.Let f(x) = 3x+6 and g(x) = x +2. Find a. (f + g) (1)b. (f – g) (2)c. (f ∙ g) (0)d. (f/g) (-1)e. (g/f) (-1)

The Composition of Functions

The composition of the function f with g is denoted by f o g and is defined by the equation

(f o g)(x) = f (g(x)).

The domain of the composite function f o g is the set of all x such that x is in the domain of g and g(x) is in the domain of f.

Example 5a.Given f (x) = 3x – 4 and g(x) = x2 + 6, find: a. (f ○ g)(x) b. (g ○ f)(x)

Example 5b.Given f (x) = x – 2 and g(x) = x + 7, find: a. (f ○ g)(x)

b. (g ○ f)(x) c. (f ○ f)(x) d. (g ○ g)(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 Testo 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. 8

6

4

2

-2

-4

5 10 15

Example 6b. Determine if the

graph is a graph of a function or just a graph of a relation.

Example 6c. Determine if the

graph is a graph of a function or just a graph of a relation.

Example 6d.16

14

12

10

8

6

4

2

2

4

6

8

15 10 5 5 10 15 20 25

A

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

Example 6e. Determine if the graph is a graph of a function or

just a graph of a relation. 4

3

2

1

-1

-2

-3

-4

-6 -4 -2 2 4 6

Example 6f.

6

4

2

-2

-4

-6

-10 -5 5 10

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

Example 6g. Determine if the graph is a graph of a function or

just a graph of a relation.

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

Finding Domain and Range from Graphs

Example 7a. : y = 2x - 4

Domain:

Range:

Example 7b.: y = x2 - 1

Domain:

Range:

Types of Functions

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

Linear Functions

Equation of a LineGeneral Form ax + by + c =

0wherein a, b, and c are real numbers

Slope-Intercept Form

y = mx + b wherein m is the slope of the line, and b is the y-intercept

Point-Slope Form

y – y1 = m (x – x1)

wherein m is the slope of the line, and P(x1, y1) is a point on the line

Example 1: Find an equation for the line through (-2,

5) and slope -3. Solution:

11 xxmyy )2(35 xy

635 xy13 xy

Example 2: Find the equation of the line through the

given pair of points (3,5) and (4,7). Solution: Find the slope Use the slope and one point, say (3, 5) in

the point-slope form

23457

mslope

11 xxmyy 325 xy

625 xy12 xy

EXERCISES1. What is the slope of the line y + 4x 1 = 0?2. What is the y-intercept of the line x + 3y = 7?3. Find the slope and the y-intercept of 2x + 4y 3

= 0.4. Find the slope of the line that passes through

(1,5) and (3,5).5. What is an equation of the line through (2,1) and

(1,4) ?

Example 3:Given: y – 2x + 6 = 0 a. Rewrite the equation in slope-intercept formb. Determine the slope.c. Find the intercepts (x and y).d. Graph the equation. Solution:

slope = 2intercepts: (0, 6) and (3, 0)

062 xy62 xy

EXERCISES For each of the given equations, do the following: Rewrite the equation in slope-intercept form Determine the slope. Find the intercepts (x and y). Graph the equation.

1. y – 2x – 8 = 02. y x + 1 = 03. 4y + 3x + 12 = 04. x – y – 6 = 0

More Exercises: Page 27 # 1 – 12 Activity Sheet 1.1 pages 353 – 356 # 1, 2,

6, 7, 8 Activity Sheet 1.2 pages 357 – 359 # 4, 5,

6 Activity Sheet 1.3 pages 361 – 362 # 1, 2,

3, 4