Functions §3.6
Fall 2013 - Math 1010
(Math 1010)
M 1010 §3.6
1 / 13
Roadmap
I
§3.6 - Functions: Relations, Functions
I
§3.6 - Evaluating Functions, Finding Domains and Ranges
(Math 1010)
M 1010 §3.6
2 / 13
§3.6 - Functions
Our study of functions will come in parts: defining a function, seeing why
functions stand out from all relations, using the notation of functions, and
finding associated domains and ranges of functions.
On 09/23 we described what parts we’ve taken from graphing. Here is a
sample of popular choices: points / order pairs / (x, y ); intercepts; shapes
- lines, curves, V -shapes; trends; axes; distance; slope between two points
Many of the above ideas are useful when learning about functions.
(Math 1010)
M 1010 §3.6
3 / 13
§3.6 - Points / Ordered Pairs
Thinking about points is a good place to start. Here is a relation table:
x
0
1
2
3
0
(Math 1010)
y
1
3
5
5
3
point
(0,1)
(1,3)
(2,5)
(3,5)
(0,3)
M 1010 §3.6
4 / 13
§3.6 - Points / Ordered Pairs
Thinking about points is a good place to start. Here is a relation table:
x
0
1
2
3
0
y
1
3
5
5
3
point
(0,1)
(1,3)
(2,5)
(3,5)
(0,3)
The set of x values is the domain. The set of y values is the range.
Domain = {0, 1, 2, 3}, Range = {1, 3, 5}
(Math 1010)
M 1010 §3.6
4 / 13
§3.6 - Points / Ordered Pairs
Thinking about points is a good place to start. Here is a relation table:
x
0
1
2
3
0
y
1
3
5
5
3
point
(0,1)
(1,3)
(2,5)
(3,5)
(0,3)
The set of x values is the domain. The set of y values is the range.
Domain = {0, 1, 2, 3}, Range = {1, 3, 5}
This relation cannot be a function.
(Math 1010)
M 1010 §3.6
4 / 13
§3.6 - Functions
A function is a rule that pairs each element of a domain to exactly one
element of the range.
(Math 1010)
M 1010 §3.6
5 / 13
§3.6 - Functions
A function is a rule that pairs each element of a domain to exactly one
element of the range.
Note:
I
No two ordered pairs have the same first component and different
second components. That is, (2, 4) and (2, 18) cannot be ordered
pairs of a function.
(Math 1010)
M 1010 §3.6
5 / 13
§3.6 - Functions
A function is a rule that pairs each element of a domain to exactly one
element of the range.
Note:
I
No two ordered pairs have the same first component and different
second components. That is, (2, 4) and (2, 18) cannot be ordered
pairs of a function.
I
The rule is a process that turns one number into another number. For
instance, ”the squares of all real numbers” is the rule that turns 1
into 1, 1.1 into 1.21, 1.2 into 1.44, and so on.
(Math 1010)
M 1010 §3.6
5 / 13
§3.6 - Functions
A function is a rule that pairs each element of a domain to exactly one
element of the range.
Note:
I
No two ordered pairs have the same first component and different
second components. That is, (2, 4) and (2, 18) cannot be ordered
pairs of a function.
I
The rule is a process that turns one number into another number. For
instance, ”the squares of all real numbers” is the rule that turns 1
into 1, 1.1 into 1.21, 1.2 into 1.44, and so on.
I
It is okay to have two or more ordered pairs with different first
components and the same second component. That is, (2, 6) and
(40, 6) and (−19, 6) may be ordered pairs of a function.
(Math 1010)
M 1010 §3.6
5 / 13
§3.6 - Functions
A function is a rule that pairs each element of a domain to exactly one
element of the range.
Algebraically this is represented with an equation. For the examples below
take x as the independent variable and y as the dependent variable.
(Math 1010)
M 1010 §3.6
6 / 13
§3.6 - Functions
A function is a rule that pairs each element of a domain to exactly one
element of the range.
Algebraically this is represented with an equation. For the examples below
take x as the independent variable and y as the dependent variable.
1) Is y 2 − x = 0 a function?
(Math 1010)
M 1010 §3.6
6 / 13
§3.6 - Functions
A function is a rule that pairs each element of a domain to exactly one
element of the range.
Algebraically this is represented with an equation. For the examples below
take x as the independent variable and y as the dependent variable.
1) Is y 2 − x = 0 a function?
2) Is x 2 − y = 0 a function?
(Math 1010)
M 1010 §3.6
6 / 13
§3.6 - Functions
A function is a rule that pairs each element of a domain to exactly one
element of the range.
Algebraically this is represented with an equation. For the examples below
take x as the independent variable and y as the dependent variable.
1) Is y 2 − x = 0 a function?
2) Is x 2 − y = 0 a function?
3) Is −4x + 8y = 5 a function?
(Math 1010)
M 1010 §3.6
6 / 13
§3.6 - Functions
A function is a rule that pairs each element of a domain to exactly one
element of the range.
Algebraically this is represented with an equation. For the examples below
take x as the independent variable and y as the dependent variable.
1) Is y 2 − x = 0 a function?
2) Is x 2 − y = 0 a function?
3) Is −4x + 8y = 5 a function?
1) No, 2) Yes, 3) Yes
(Math 1010)
M 1010 §3.6
6 / 13
§3.6 - Function Notation
When writing the rule of a function in equation form, it is convenient to
keep track of input and output pairs and to give the function a name.
This is all done with function notation.
(Math 1010)
M 1010 §3.6
7 / 13
§3.6 - Function Notation
When writing the rule of a function in equation form, it is convenient to
keep track of input and output pairs and to give the function a name.
This is all done with function notation.
Example: y = (x + 3)3 becomes f (x) = (x + 3)3
(Math 1010)
M 1010 §3.6
7 / 13
§3.6 - Function Notation
When writing the rule of a function in equation form, it is convenient to
keep track of input and output pairs and to give the function a name.
This is all done with function notation.
Example: y = (x + 3)3 becomes f (x) = (x + 3)3
• The name of the function is f .
• f (x) is read, ”f-of-x,” or, ”the value of f at x.”
Evaluating a function is done by subsituting an x value into the function.
Example: Evaluate the value of f (x) = (x + 3)3 at x = −5.
(Math 1010)
M 1010 §3.6
7 / 13
§3.6 - Function Notation
When writing the rule of a function in equation form, it is convenient to
keep track of input and output pairs and to give the function a name.
This is all done with function notation.
Example: y = (x + 3)3 becomes f (x) = (x + 3)3
• The name of the function is f .
• f (x) is read, ”f-of-x,” or, ”the value of f at x.”
Evaluating a function is done by subsituting an x value into the function.
Example: Evaluate the value of f (x) = (x + 3)3 at x = −5.
f (−5) = ((−5) + 3)3 = (−2)2 = −8.
(Math 1010)
M 1010 §3.6
7 / 13
§3.6 Functions
Are some values of x not allowed for
f (x) =
(Math 1010)
1
?
x − 10
M 1010 §3.6
8 / 13
§3.6 Functions
Are some values of x not allowed for
f (x) =
1
?
x − 10
Are some values of x not allowed for
f (x) =
(Math 1010)
√
x?
M 1010 §3.6
8 / 13
§3.6 - Implied Domains and Ranges
Implied domains are the set of real values for x that yield real values for
f (x). The rule for the function implies the domain. Exclude:
I
sets of values for x that result in division by zero.
I
sets of values for x that result in taking the square-roots of negative
numbers.
Otherwise, explicit domains very directly state which x values to exclude
when the rule of the function is given.
Example: f (x) = x − 7, x 6= 7. The domain is {x|x 6= 7} or ’all x except
7.’
(Math 1010)
M 1010 §3.6
9 / 13
§3.6 Implied Domain and Range
Find the implied domain of
f (x) =
(Math 1010)
√
5x − 2
M 1010 §3.6
10 / 13
§3.6 Implied Domain and Range
Find the implied domain of
f (x) =
√
5x − 2
The inequality is 5x − 2 ≥ 0
(Math 1010)
M 1010 §3.6
10 / 13
§3.6 Implied Domain and Range
Find the implied domain of
f (x) =
√
5x − 2
The inequality is 5x − 2 ≥ 0
Solved: x ≥ 25 . All real numbers x such that x ≥
(Math 1010)
M 1010 §3.6
2
5
is the domain.
10 / 13
§3.6 Explicit Domain and Range
The domain of f given below is all real numbers. What is the rule?
(
−x, if x ≤ 0
f (x) =
x,
if x > 0
(Math 1010)
M 1010 §3.6
11 / 13
§3.6 Explicit Domain and Range
The domain of f given below is all real numbers. What is the rule?
(
x + 8, if x < 2
f (x) =
6 − 3x, if x ≥ 2
(Math 1010)
M 1010 §3.6
12 / 13
Assignment
Assignment:
For Wenesday:
1. Exercises from §3.6 due Wednesday, October 2.
2. Exam # 1: Chapter 3 & Cumulative Chapters 1 - 2, October 2
(Math 1010)
M 1010 §3.6
13 / 13