Algebra 1
Unit 3
Functions
Rate of Change
Rate of change- a rate that describes how one quantity changes in relation to another
quantity.
Patterns
Student Practice
Patterns Partner Activity
For each pattern, see if you can find a rule for the pattern and write it as an equation.
The first 2 are done for you as examples.
1. (0, 1), (1, 3), (2, 5), (3, 7), (4, 9)
2.
y=x-3
3. (0, 0), (1, 1), (2, 4), (3, 9), (4, 16)
5. (0, 0), (1, 1), (2, 8), (3, 27), (4, 64)
5. (0, 1), (1, 5), (2, 9), (3, 13), (4, 17)
6. (0, 21), (1, 0), (2, 7), (3, 26), (4, 63)
7. (0, 2), (1, 1), (2, 0), (3, 21), (4, 22)
8. (2, 5), (4, 9), (5, 11), (7, 15), (10, 21)
9. (2, 5), (3, 10), (4, 17), (5, 26), (6, 37)
10.
y = 2x + 1
Writing Function Rules
For Exercises 1–10, write a function rule that represents each situation.
1. The price p of an ice cream is $3.95 plus $0.85 for each topping t on the ice
cream.
2. A babysitter’s earnings e are a function of the number of hours n worked at a rate
of $7.25 per hour.
3. The price p of a club’s membership is $30 for an enrollment fee and $12 per
week w to be a member.
4. A plumber’s fees f are $75 for a house call and $60 per hour h for each hour
worked.
5. A hot dog d costs $1 more than one-half the cost of a hamburger h.
6. José is 3 years younger than 3 times his brother’s age. Write a rule that
represents José’s age j as a function of his brother’s age b. How old is José if his
brother is 5?
7. A taxicab charges $4.25 for the first mile and $1.50 for each additional mile.
Write a rule for describing the total rate r as a function of the total miles m. What
is the taxi rate for 12 miles?
8. The price p of a large, cheese pizza is $7.95 plus $0.75 for each topping t on the
pizza.
9. Jaquelin’s earnings m are a function of the number of lawns n she mows at a
rate of $12 per lawn.
10. The total fees f of a book club membership are $10 per month m and a onetime administrative fee of $4.75.
Introduction to Functions
Mapping Diagrams
Relations and functions can also be expressed as a correspondence or mapping from
one set to another.
In the example below of function F, the arrow from 1 to 2 indicates that the ordered pair
(1, 2) belongs to function F. Each first component is paired with exactly one second
component, therefore, the mapping diagram represents a function!
X values
1
–2
3
y values
2
4
–1
In the mapping below, the input –2 is paired with two different output values, 1 and 0,
therefore, it is not a function!!
input
output
–4
–2
1
0
Domain and Range
Example 3: Give the domain and range of the relation. Tell whether the relation defines a
function.
x
-5
0
5
y
2
2
2
The domain is {____________________}
The range is {_____________________}
Is it a function?
Example 4: Give the domain and range
of the relation.
Example 5: Give the domain and range
of the relation.
Tell whether the relation defines a function.
Tell whether the relation defines a function.
The domain is {____________________}
The domain is {_________________}
The range is {_____________________}
The range is {__________________}
Is it a function?
Is it a function?
These are all examples of discrete functions. What does that mean?????
Continuous
Discrete
Tell whether each graph is discrete or continuous.

When we find domain and range of discrete functions, it is written as a list of
numbers and we think of domain as all the x values, range is all the y-values!!!!

When we find domain and range of continuous functions, we use different
notation (we will discuss later)
Student Practice
13.
14.
15.
Draw a graph with the given domain and range.
Use the Vertical Line Test to determine if the relation is a function.
Domain and Range Notation ~ Continuous Functions


In some cases, we must use interval or set-builder notation when writing the
domain and range of a function because the number of values of the
domain/range is infinite (____________________________________________).
Think of domain as “left to right” and range as “low to high”.

In interval notation, there are only 4 symbols to know:
*Open parentheses ( )
*Closed parentheses [ ]
*Infinity ∞

*Negative Infinity −∞
With inequality notation, the symbols we use are <, >, <, and >.
To use interval notation:

Use the open parentheses ( ) if the value is not included in the graph. (i.e. the
graph is undefined at that point... there's a hole or asymptote, or a jump)

If the graph goes on forever to the left, the domain will start with ( −∞. If the graph
travels downward forever, the range will start with ( −∞. Similarly, if the graph
goes on forever at the right or up, end with ∞)

Use the brackets [ ] if the value is part of the graph.
To use inequality notation:

Look to see if the values are getting increasing (>) or decreasing (<)

If the point(s) are included, use < or >. If the point is not included, use < or >.
Let’s practice with this notation before we find domain and range.
Finding Domain and Range of Continuous Functions
Interval and Inequality Notation
Example 1: Give the domain and range of the relation. Tell whether the relation defines a
function.
The domain is
The range is
Is it a function?
Example 2: Give the domain and range of the relation. Tell whether the relation defines a
function.
The domain is
The range is
Is it a function?
Example 3: Give the domain and range of the relation. Tell whether the relation defines a
function.
The domain is
The range is
Is it a function?
Example 4: Give the domain and range of the relation. Tell whether the relation defines a
function.
The domain is
The range is
Is it a function?
Example 5: Give the domain and range of the relation. Tell whether the relation defines a
function.
a.
b.
The domain is
The domain is
The range is
The range is
Is it a function?
Is it a function?
Student Practice
Evaluating Functions
Using a Function
Evaluating Functions Using Graphs
Refer to the following graph. Find f(6).
Refer to the following graph. Find f(10).
Refer to the following graph. Find f(-4).
Refer to the following graph. Find f(2).
Writing Functions
1) Yawgoo Ski Resort offers an equipment rental package for $35. They also offer
new skiers lessons for $44 an hour.
Write a function model to express the cost, C of renting equipment and taking lessons
for h hours.
2) Verizon is advertising a package for new customers. The charge is $11 per month
after a $65 installation fee. Cox is competing for new customers by offering a similar
package for new customers. There is no installation fee but the monthly cost is $16.
a. Write a function that shows the cost, f for joining Verizon for m months.
b. Write a function that shows the cost, g for joining Cox for m months.
c. Which company offers a better price for a year and a half?
Student Practice
1)
2)
3)
4) Let f(x) = 2x – 1. Find each of the following.
a. f(x + 3)
b. f(4x)
c. f(2x – 1)
6. Refer to the following graph to find the following:
a. f(0)
b. f(3)
c. f(5)
d. f(-4)