The document provides an outline for a lesson on functions and relations. It includes: - A review of functions as machines, tables of values, graphs, and the vertical line test. - How functions can represent real-life situations, including piecewise functions. - An example of using a piecewise function to model the temperature of water as heat is added. - The lesson aims to represent real-life situations using functions and solve problems involving functions.

1. FUNCTIONS AND
RELATIONS
Chapter 1

2. FUNCTIONS AS
MODELS
At the end of this lesson, you should be able to represent real life situations using
functions, including piecewise functions, and solve problems involving functions.

3. Lesson Outline
■ Review: of relations and Functions
■ Review:The function as a machine
■ Review: Functions and relations as a table of values
■ Review: Functions as a graph in the Cartesian plane
■ Review:Vertical LineTest
■ Functions as representations of real life situations
■ Piecewise Functions

4. Functions and Relations

5. Functionasamachine

6. Functions and relations as a table of
values

7. Functions and relations as a table of
values
one to one many to one one to many

8. Functions as a graph in the Cartesian
plane

9. Vertical LineTest
Example 3.Which of the following graphs can be graphs of
functions?

10. Vertical LineTest
Example 3.Which of the following graphs can be graphs of
functions?

11. Checkpoint:
■When is a relation said to be a
function?
■Explain the vertical line test

12. Group Activity: 5 minutes
■Answer the given
worksheet in 5 minutes
5 432 1

14. Which of the following represents a
function?
a) y=2x+1
b) y= x2 -2x+2
c) x2 + y2 =1
d) y= 𝑥 + 1
e) y=
2𝑥+1
𝑥−1

15. The domain of a relation is the set of all
possible values that the variable x can take
a) y=2x+1
b) y= x2 -2x+2
c) x2 + y2 =1
d) y= 𝑥 + 1
e) y=
2𝑥+1
𝑥−1

16. If a relation is a function, then y can be
replaced with f(x) to denote that the value of
y depends on the value of x.
Replace y in the following examples to denote a
function:
a) y=2x+1
b) y= x2 -2x+2
c) y= 𝑥 + 1
d) y=
2𝑥+1
𝑥−1

17. If a relation is a function, then y can be
replaced with f(x) to denote that the value of
y depends on the value of x.
Replace y in the following examples to denote a
function:
a) f(x)=2x+1
b) q(x)= x2 -2x+2
c) g(x)= 𝑥 + 1
d) r(x)=
2𝑥+1
𝑥−1

18. Checkpoint:
■ Define function
■ When is a relation said to be a function?
■ What is the use of the vertical line test?

19. Important concepts
■ Relations are rules that relate two values, one
from a set of inputs and the second from the set
of outputs.
■ Functions are rules that relate only one value
from the set of outputs to a value from the set
of inputs.

20. Quiz: Determine whether each of the
following relation is a function or not
1. {(1,-2), (-2,0), (-1,2), (1,3)}
2. {(1,1), (2,2), (3,5), (4,10), (5,15)}
3. y2= 3x+2
4. y=4-5x
5.
6.
7. 8.

21. Quiz: Determine whether each of the
following relation is a function or not
9. 10.

22. FUNCTIONS AS
REPRESENTATIONS OF
REAL-LIFE SITUATIONS
What are piecewise functions?

23. Using functions in real-life situations

24. One hundred meters of fencing is available to enclose a
rectangular area next to a river. Give a function A that
can represent the area that can be enclosed, in terms of
x.

25. PIECEWISE
FUNCTIONS
Some situations can only be described
by more than one formula

26. A user is charged P300 monthly for a
particular mobile plan, which includes 100
free text messages. Messages in excess of
100 are charged P1 each. Represent the
monthly cost for text messaging using the
function t(m), where m is the number of
messages sent in a month.

27. A jeepney ride costs P8.00 for the first 4 kilometers, and
each additional integer kilometer adds P1.50 to the fare.
Use a piecewise function to represent the jeepney fare
in terms of the distance (d) in kilometers.

28. Water can exist in three states: solid ice, liquid water, and gaseous water
vapor. As ice is heated, its temperature rises until it hits the melting point
of 0°C and stays constant until the ice melts.The temperature then rises
until it hits the boiling point of 100°C and stays constant until the water
evaporates. When the water is in a gaseous state, its temperature can rise
above 100°C (This is why steam can cause third degree burns!). A solid
block of ice is at -25°C and heat is added until it completely turns into water
vapor. Sketch the graph of the function representing the temperature of
water as a function of the amount of heat added in Joules given the
following information:
The ice reaches 0°C after applying 940 J.
The ice completely melts into liquid water after applying a total of 6,950
J.
The water starts to boil (100°C) after a total of 14,470 J.
The water completely evaporates into steam after a total of 55,260 J.
Assume that rising temperature is linear. Explain why this is a piecewise
function.

29. Solution. LetT(x) represent the temperature of the
water in degrees Celsius as a function of cumulative heat
added in Joules.The functionT(x) can be graphed as
follows:

30. EVALUATING
FUNCTIONS
At the end of this lesson, you will be able to evaluate functions

31. Evaluating a function means replacing
the variable in the function.

32. Solution: Substitute 1.5 for x

33. Seatwork