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 whyfunctions stand out from all relations, using the notation of functions, andfinding associated domains and ranges of functions.

On 09/23 we described what parts we’ve taken from graphing. Here is asample 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 y point

0 1 (0,1)1 3 (1,3)2 5 (2,5)3 5 (3,5)0 3 (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 - Points / Ordered Pairs

Thinking about points is a good place to start. Here is a relation table:

x y point

0 1 (0,1)1 3 (1,3)2 5 (2,5)3 5 (3,5)0 3 (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 - Points / Ordered Pairs

Thinking about points is a good place to start. Here is a relation table:

x y point

0 1 (0,1)1 3 (1,3)2 5 (2,5)3 5 (3,5)0 3 (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 oneelement of the range.

Note:

I No two ordered pairs have the same first component and differentsecond components. That is, (2, 4) and (2, 18) cannot be orderedpairs of a function.

I The rule is a process that turns one number into another number. Forinstance, ”the squares of all real numbers” is the rule that turns 1into 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 firstcomponents 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 oneelement of the range.

Note:

I No two ordered pairs have the same first component and differentsecond components. That is, (2, 4) and (2, 18) cannot be orderedpairs of a function.

I The rule is a process that turns one number into another number. Forinstance, ”the squares of all real numbers” is the rule that turns 1into 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 firstcomponents 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 oneelement of the range.

Note:

I No two ordered pairs have the same first component and differentsecond components. That is, (2, 4) and (2, 18) cannot be orderedpairs of a function.

I The rule is a process that turns one number into another number. Forinstance, ”the squares of all real numbers” is the rule that turns 1into 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 firstcomponents 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 oneelement of the range.

Note:

I No two ordered pairs have the same first component and differentsecond components. That is, (2, 4) and (2, 18) cannot be orderedpairs of a function.

I The rule is a process that turns one number into another number. Forinstance, ”the squares of all real numbers” is the rule that turns 1into 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 firstcomponents 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 oneelement of the range.

Algebraically this is represented with an equation. For the examples belowtake x as the independent variable and y as the dependent variable.

1) Is y2 − x = 0 a function?

2) Is x2 − 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 - Functions

A function is a rule that pairs each element of a domain to exactly oneelement of the range.

Algebraically this is represented with an equation. For the examples belowtake x as the independent variable and y as the dependent variable.

1) Is y2 − x = 0 a function?

2) Is x2 − 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 - Functions

A function is a rule that pairs each element of a domain to exactly oneelement of the range.

Algebraically this is represented with an equation. For the examples belowtake x as the independent variable and y as the dependent variable.

1) Is y2 − x = 0 a function?

2) Is x2 − 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 - Functions

A function is a rule that pairs each element of a domain to exactly oneelement of the range.

Algebraically this is represented with an equation. For the examples belowtake x as the independent variable and y as the dependent variable.

1) Is y2 − x = 0 a function?

2) Is x2 − 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 - Functions

A function is a rule that pairs each element of a domain to exactly oneelement of the range.

Algebraically this is represented with an equation. For the examples belowtake x as the independent variable and y as the dependent variable.

1) Is y2 − x = 0 a function?

2) Is x2 − 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 tokeep 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 - Function Notation

When writing the rule of a function in equation form, it is convenient tokeep 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 - Function Notation

When writing the rule of a function in equation form, it is convenient tokeep 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 - Function Notation

When writing the rule of a function in equation form, it is convenient tokeep 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) =1

x − 10?

Are some values of x not allowed for

f (x) =√x ?

(Math 1010) 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) =√x ?

(Math 1010) 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 forf (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 negativenumbers.

Otherwise, explicit domains very directly state which x values to excludewhen 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 except7.’

(Math 1010) M 1010 §3.6 9 / 13

§3.6 Implied Domain and Range

Find the implied domain of

f (x) =√

5x − 2

The inequality is 5x − 2 ≥ 0Solved: x ≥ 2

5 . All real numbers x such that x ≥ 25 is the domain.

(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 ≥ 2

5 is the domain.

(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 ≥ 0Solved: x ≥ 2

5 . All real numbers x such that x ≥ 25 is the domain.

(Math 1010) M 1010 §3.6 10 / 13

§3.6 Explicit Domain and Range

The domain of f given below is all real numbers. What is the rule?

f (x) =

{−x , if x ≤ 0

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?

f (x) =

{x + 8, if x < 2

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