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
12. Group Activity: 5 minutes
■Answer the given
worksheet in 5 minutes
5 432 1
13.
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
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.
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.
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:
Which of these machines, if you know the input, can you determine a single or unique output? answer: a, c, d, f. Why do b and e not included?
Which of these machines, if you know the output, can you determine a single or unique input? answer: d, f. why? This will be used in the discussion of one-to-one functions
Suppose we connect machine a to machine c, such that the output of a becomes the input of c. give the outputs for S, U, G, A, R
The relations f and h are functions because no two ordered pairs have the same x-value but different y-values, while g is not a function because (1,3) and (1,4) are ordered pairs with the same x-value but different y-values.
The relations f and g are functions because each xEX corresponds to a unique yEY. The relation h is not a function because there is at least on element in X for which there is more than one corresponding y-value. For example, x=7 corresponds to y=11 or 13. similarly, x=2 corresponds to both y=17 or 19
Recall from junior high school (G8) that a relation between two sets of numbers can be illustrated by a graph in the Cartesian plane, and that a function passes a VERTICAL LINE TEST
By convention , the x-variable is the input variable and that the value of the y-variable is computed based on the value of the x-variable. A relation is a function if for each x-value there corresponds only one y-value
All are relations. All are functions except c. equation c is not a function because can find an x-value that corresponds to more that one y-value.
A-all real numbers
B-all real numbers
C- [-1,1]
D- [-1 to positive infinity)
E- (negative infinity to 1)union(1 to positive infinity)