COLLEGEALGEBRA
Lesson: Functions and Their Graphs
Objectives:
1. Determine whether a relation represents a function;
2. Evaluate a function for a given input;
3. Find the domain of a function
4. Operate with functions
5. Interpret the graph of a function
Part 1: Relations and Functions
A relation is a correspondence between two sets.
If x and y are two elements in these sets and if a relation exists between x and y, then we say that
x corresponds to y or that y depends on x, and we write x  y.
A relation may be represented as a diagram, using set-notation or by an equation/expression.
Ex 1:
Diagram
State:
Arizona
California
Florida
New York
South Carolina
Texas
No of representatives
7
9
27
36
53
Set notation: {(Arizona,9), (California,53), (Florida,27), (New York,27),(South Carolina,7),
(Texas,36)}.
Ex 2:
Diagram
Animal:
Dog
Duck
Rabbit
Set notation: {(Dog,11), (Duck,10), (Rabbit,7)}.
Life
Expectancy:
7
10
11
COLLEGEALGEBRA
If X and Y are two non-empty sets, then a function from X to Y is a relation that associates with
each element of X exactly one element of Y.
The set X is called the domain of the function.
The elements of the set X are called inputs, and the corresponding elements in Y are called
outputs. The domain is also the set of all the inputs, and the set of all the outputs is called the
range of the function.
The standard notation for a function is:
:
→ ,
.
The Examples 1 and 2 above are functions.
Ex 3: Determine whether each relation represents a function. If it is a function, state the domain
and range:
a. {(2, 3), (4, 1), (3, -2), (2, -1)}
Not a function;
b. {(-2, 3), (4, 1), (3, -2), (2, -1)}
Yes. Domain = {-2,2,3,4}, Range = {-2,-1,1,3};
c. {(2, 2), (4, 3), (3, 3), (8, -1)}
Yes. Domain = {2,3,4,8}, Range = {-1,2,3}.
Ex 4: Determine if the equation
2
3 defines y as a function of x.
For each value we choose for the input x, there is only one value for y. The answer is yes.
Ex 5: Determine if the equation
The expression of y is:
The answer is no.
2
3 defines y as a function of x.
. For the value x = 5, we get two different values for y: 1.
COLLEGEALGEBRA
Part 2: Evaluating a Function
Ex 6: If
3
1, find the value
2 .
We evaluate the function for the value of the input x = -2:
2
3 2
1
6 1
5, so the output is -5.
Ex 7: If
, find the value
1 .
We evaluate the function for the value of the input x = -1:
0, so the output is 0.
1
Part 3: The Domain of a Function
If
then we set
If
then we have no restrictions on
then we set
If
0 and solve it. The solution set will give the domain.
.
0 and solve it. The domain will consist in all real numbers
except for the solutions of that equation.
Ex. 8:
We set
1,4 .
3
Ex. 9:
We set 2
5.
√2
5 0 and solve it:
4
or interval notation
0 and solve it:
4,
1. So the domain is R\{-1,4}, or
⋮
. The domain can be expressed in set notation
⋮
, .
Ex. 10:
2
3
7.
This is a polynomial function and we have no restriction regarding the inputs, so that the domain
is R.
COLLEGEALGEBRA
Part 4: Operations with functions
Sum of f + g
Difference of f - g
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
Product of f g
Quotient of f/ g
(f g)(x) = f(x)g(x)
(f /g)(x) = f(x)/g(x)
,
Ex 11:
.
a). the domain of the function f: R\{0}.
b). the domain of function g: R\{-1}.
c).
.
d).
e).
.
∙
∙
f).
∙
∙
.
.
Part 5: The Graph of a function
Vertical Line Test: A set of points in a xy-plane is the graph of a function if and only if every
vertical line intersects the graph in at most one point.
Ex 12: Functions
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Ex 13: Not functions
Ex 14:
a. What are
0 ,
,
3
?
From the graph, 0
4,
0,
b. The domain of f is 0,4 .
c. The range of f is [-4,4].
d. The y-intercept is (0,4).
The x-intercepts are: , 0 ,
3
4.
,0 ,
e. The function is positive (f(x)>0) for
Ex 15:
,0 ,
∈ 0,
,0 .
∪
,
∪
,4
.
.
a. Domain of the function: because the equation
1 0 has the solution x=1,
then the domain will be ⋮
1 , or R\{1}.
b. Is the point (2,2) on the graph?
If x=2 is replaced, then we get 2
2, so that the point (2,2) is on
the graph.
c. If x=4, then 4
, so that the point 4, is on the graph of f.
d. If f(x)=5, then
5→
, so the point
, 5 is on the graph of f.
COLLEGEALGEBRA
Part 6: The Difference Quotient
If f(x) is a function, then the expression
,
0.
is called the difference quotient. It has applications in Calculus, where is introducing the
concept of derivative.
3
Ex 16: The difference quotient for
5 is:
5
3
3
3
2
6
3
6
5
5
5
5
3
6
6
3
3
3
5
5
5.
3
3
5
5
5