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