1. Grab a set of interactive notes Guide due: A Day 10/24 B Day 10/25
Section 3: Introduction to Functions Topic 1-2
Topics 1
Welcome!
Grab a set of interactive notes
Guide due: A Day 10/24 B Day 10/25
Topics 2

2. Section 3: Introduction to Functions Topic 1 - 2
Students will:
Identify functions. Find the domain and range of relations and functions.
Identify independent and dependent variables. Write an equation in function notation and evaluate a function for given input values.

3. Warm Up Section 3: Introduction to Functions Topic 1 - 2
Evaluate each expression for a = 2, b = –3, and c = 8.
1. a + 3c
2. ab – c 3. 26
–14 1 2
c + b 1
4. 4c – b 35
5. ba + c 17

4. Section 3: Functions Topic 1 – Input and Output Values
What is a Function?
A function relates an input to an output.
We will see many ways to think about functions, but there are always three main parts:
The input
The relationship
The output

5. A function relates an input to an output.
Section 3: Functions Topic 1 – Input and Output Values
Example: "Multiply by 2" is a very simple function.
Input
Relationship
Output 1 2 4 3 9 16 5
A function relates an input to an output.
For an input of 5, what is the output?

6. Section 3: Introduction to Functions Topic 1 - 2
Names
It is useful to give a function a name.
The most common name is "f", but we can have other names like "g“
It is like a machine that has an input and an output.
And the output is related somehow to the input.

7. We say "f of x equals x squared"
Section 3: Introduction to Functions Topic 1 - 2
We say "f of x
equals x squared"
What goes into the function is put inside parentheses ( ):
So f(x) shows us the function is called "f", and "x" goes in
And we usually see what a function does with the input:
f(x) = x2 shows us that function "f" takes "x" and squares it.

8. Section 3: Introduction to Functions Topic 1 - 2
Example: with f(x) = x2:
An input of 4 simply becomes an output of 16.
In fact we can write f(4) = 16.
The "x" is Just a Place-Holder for the input!
Let’s look at other examples

9. Example 4A: Evaluating Functions
Section 3: Introduction to Functions Topic 1 - 2
Example 4A: Evaluating Functions
Evaluate the function for the given input values.
For f(x) = 3x + 2, find f(x) when x = 7 and when x = –4.
f(x) = 3(x) + 2
f(x) = 3(x) + 2
Substitute
7 for x.
Substitute
–4 for x.
f(7) = 3(7) + 2
f(–4) = 3(–4) + 2 =
Simplify.
Simplify.
= –12 + 2
= 23
= –10

10. Example 4B: Evaluating Functions
Section 3: Introduction to Functions Topic 1 - 2
Example 4B: Evaluating Functions
Evaluate the function for the given input values.
For g(t) = 1.5t – 5, find g(t) when t = 6 and when t = –2.
g(t) = 1.5t – 5
g(t) = 1.5t – 5
g(6) = 1.5(6) – 5
g(–2) = 1.5(–2) – 5
= 9 – 5
= –3 – 5
= 4
= –8

11. Example 4C: Evaluating Functions
Section 3: Introduction to Functions Topic 1 - 2
Example 4C: Evaluating Functions
Evaluate the function for the given input values.
For , find h(r) when r = 600 and when r = –12.
= 202
= –2

12. Section 3: Introduction to Functions Topic 1 - 2
Check It Out! Example 4a
Evaluate the function for the given input values.
For h(c) = 2c – 1, find h(c) when c = 1 and when c = –3.
h(c) = 2c – 1
h(c) = 2c – 1
h(1) = 2(1) – 1
h(–3) = 2(–3) – 1
= 2 – 1
= –6 – 1
= 1
= –7

13. Section 3: Introduction to Functions Topic 1 - 2
Formal Definition of a Function:
A function relates each element of a set with exactly one element of another set.
A function is a type of relation where each x value (domain) can be paired with only one y value (range).

14. Section 3: Introduction to Functions Topic 1 - 2
Let’s explore:
Each x value can be paired with only one y value

15. Section 3: Introduction to Functions Topic 1 - 2
The domain of a relation is the set of first coordinates (or x-values) of the ordered pairs.
The range of a relation is the set of second coordinates (or y-values) of the ordered pairs.

16. Domain Range y Input f(x)
Section 3: Introduction to Functions Topic 1 - 2
Other Names for Domain and Range
Domain
Range  y
Input
 f(x)

17. Section 3: Introduction to Functions Topic 1 - 2
Example 2: Finding the Domain and Range of a
Relation
Give the domain and range of the relation.
The domain value is all x-values from 1 through 5, inclusive.
The range value is all y-values from 3 through 4, inclusive.
Domain: 1 ≤ x ≤ 5
Range: 3 ≤ y ≤ 4

18. Section 3: Introduction to Functions Topic 1 - 2
Check It Out! Example 2a
Give the domain and range of the relation. 1 2 6 5
The domain values are all x-values 1, 2, 5 and 6. –4 –1
The range values are y-values 0, –1 and –4.
Domain: {6, 5, 2, 1}
Range: {–4, –1, 0}

19. x y Section 3: Introduction to Functions Topic 1 - 2
Check It Out! Example 2b
Give the domain and range of the relation.
x y 1 4 8
The domain values are all x-values 1, 4, and 8.
The range values are y-values 1 and 4.
Domain: {1, 4, 8}
Range: {1, 4}

20. Example 3A: Identifying Functions
Section 3: Introduction to Functions Topic 1 - 2
Example 3A: Identifying Functions
Give the domain and range of the relation. Tell whether the relation is a function. Explain.
{(3, –2), (5, –1), (4, 0), (3, 1)}
Even though 3 is in the domain twice, it is written only once when you are giving the domain.
D: {3, 5, 4}
R: {–2, –1, 0, 1}
The relation is not a function. Each domain value does not have exactly one range value. The domain value 3 is paired with the range values –2 and 1.

21. Example 3B: Identifying Functions
Section 3: Introduction to Functions Topic 1 - 2
Example 3B: Identifying Functions
Give the domain and range of the relation. Tell whether the relation is a function. Explain. –4
Use the arrows to determine which domain values correspond to each range value. 2 –8 1 4 5
D: {–4, –8, 4, 5}
R: {2, 1}
This relation is a function. Each domain value is paired with exactly one range value.

22. Example 3C: Identifying Functions
Section 3: Introduction to Functions Topic 1 - 2
Example 3C: Identifying Functions
Give the domain and range of the relation. Tell whether the relation is a function. Explain.
D: –5 ≤ x ≤ 3 R: –2 ≤ y ≤ 1
The relation is not a function. Nearly all domain values have more than one range value.

23. Section 3: Introduction to Functions Topic 1 - 2
The input of a function is the independent variable.
The output of a function is the dependent variable.
The value of the dependent variable depends on,
or is a function of, the value of the independent variable.

24. Example 2C: Identifying Independent and Dependent Variables
Section 3: Introduction to Functions Topic 1 - 2
Example 2C: Identifying Independent and Dependent Variables
Identify the independent and dependent variables
in the situation.
A veterinarian must weigh an animal before determining the amount of medication.
The amount of medication depends on the weight of an animal.
Dependent: amount of medication
Independent: weight of animal

25. Section 3: Introduction to Functions Topic 1 - 2
Check It Out! Example 2a
Identify the independent and dependent variable in the situation.
A company charges $10 per hour to rent a jackhammer.
The cost to rent a jackhammer depends on the length of time it is rented.
Dependent variable: cost
Independent variable: time

26. Section 3: Introduction to Functions Topic 1 - 2
Check It Out! Example 2b
Identify the independent and dependent variable in the situation.
Apples cost $0.99 per pound.
The cost of apples depends on the number of pounds bought.
Dependent variable: cost
Independent variable: pounds

27. Section 3: Introduction to Functions Topic 1 - 2
Helpful Hint
There are several different ways to describe the variables of a function.
Independent
Variable
Dependent
Variable
x-values
y-values
Domain
Range
Input
Output x
f(x)

28. Section 3: Introduction to Functions Topic 1 - 2
Homework
Topics 3 and 4