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FUNCTIONS
Lesson 1.1 Functions in Real-life
In mathematics, a function is a relationship between two sets of
elements in which no element in the first set relates to more than one element
in the second. We can think of a function as a rule that takes inputs from the
first set and relates them to an element in the second set, which is the output
(Function Application for the Real World, 2017).For example, if you take tests,
there is a function that tells you if you get this score on your test ,you will get
this grade. If you go grocery shopping, there is a function that tells you if you
buy this many bottles of water ,you need to pay this much money. If you pay
for cable TV, there is a function that tells you if you want this many channels,
you need to pay this much money (Functions: Identification, Notation &
Practice Problems,2013).
Guide Questions:
1. In the example: If you pay for cable TV, there is a function that tells you
if you want this many channels, you need to pay this much money.
What is the input variable?the output variable?
_____________________________________________________________________
2. How did you identify the input variable from the output variable?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
In addition, functions are used as representation of real-life situations
that shows relationship between the sets of values or quantities such as test
score and grade, and number of bottles of water and amount paid. The test
score does changes in the grade shows relationship, that is grade
(output/dependent variable) is dependent to test score (input/independent
variable).
We can use function notation for functions, such as f(x) = y, where:
x is the input/independent variable,
y is the output/dependent variable
f is the name of the function (we can also use letters other than f).
f(x) is the value of the function at x.

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In this lesson, you are expected to construct mathematical models to
represent real-life situations using functions.
Example 1: Give a function P that can represent the cost of buying x large
size t-shirt if each t-shirt costs P160.
Solution:
Step 1: Identify the input and output in the situation and assign
corresponding variable for each. Remember that, output always depends on
input.
➢ The input is number of large size t-shirt (x) and the output is the
cost of t-shirts (C).
Step 2: Write the function notation to be used to represent the situation.
➢ Since the cost of buying (C) depends on the number of large size
t-shirt (x), then the function notation is C(x).
Step 3: Construct a mathematical expression based on the mathematical
phrase present in the situation. This will be the subfunctions.
➢ Each t-shirt costs P160 is translated as mathematical expression
as: 160x
Step 4: Combine the function notation and mathematical expression using
(=) sign. The result will be the linear function that represent the cost (C) of
buying large size t-shirts (x).
➢ C(x) = 160x
Note: Remember, functions can also be written in different ways using other
variables such as C(x),h(x), and g(x). Similarly, functions may take other input
values other than x such as P(a), h (r), and g(m).
PIECEWISE FUNCTIONS. Some situations can only be described by more
than one formula, depending on the value of the independent variable.
Definition: A piecewise function is defined by multiple subfunctions, where
each subfunction applies to a certain interval of the main function’s domain.
Example 2: A videoke machine can be rented for P1,000 for three days, but
for the fourth day onwards, an additional cost of P400 per day is added.
Represent the cost of renting a videoke machine as a piecewise function of the
number of days it is rented.

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Solution:
Step 1: Identify the input and output in the situation and assign
corresponding variable for each. Remember that, output always depends on
input.
➢ The input is the number of days (x) rented and the output is the
cost of renting (r).
Step 2: Write the function notation to be used to represent the situation.
➢ Since the cost of renting (r) depends on number of days (x) rented,
then the function notation is r(x).
Step 3: Construct a mathematical expression based on the mathematical
phrase present in the situation. This will be the subfunctions.
➢ rented for P1,000 for three days is translated as mathematical
expression as:1000,𝑖𝑓 0 ≤ 𝑥 ≤ 3
➢ an additional cost of P400 per day is added to 1000 is translated
as mathematical expression as: 1000 + 400(𝑥 − 3) , 𝑖𝑓 𝑥 > 3
Step 4: Combine the function notation and subfunctions using equal (=) sign.
The result will be the piecewise function that represent the cost of renting (r)
in terms of number of days (x).
➢ 𝑟(𝑥) = {
1000 , 𝑖𝑓 0 ≤ 𝑥 ≤ 3
1000 + 400(𝑥 − 3) , 𝑖𝑓 𝑥 > 3
Function Notation Subfunctions Intervals
r(x) 1000
1000 + 400(x – 3)
0 ≤ 𝑥 ≤ 3
𝑥 > 3
Notes:
• 0 ≤ 𝑥 ≤ 3 and 𝑥 > 3 are intervals of the piecewise function based on the
given information in the problem.
• fourth day onwards means four days and up
• additional cost means increased in cost (addition).
Example 3: The cost of hiring a catering service to serve food for a party is
P150 per head for 20 persons or less, P130 per head for 21 to 50 persons, and
P110 per head for 51 to 100 persons. For more than 100 persons, the cost is
at P100 per head. Represent the total cost as a piecewise function of the
number of attendees of the party.
Solution:
Step 1: Identify the input and output in the situation and assign
corresponding variable for each. Remember that, output always depends on
input.

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➢ The input is the number of attendees/persons (x) and the output
is the cost of hiring a catering service (f).
Step 2: Write the function notation to be used to represent the situation.
➢ Since the cost of hiring a catering services (f) depends on the
number of attendees/persons (x), then the function notation is
f(x).
Step 3: Construct a mathematical expression based on the mathematical
phrase present in the situation. This will be the subfunctions.
➢ P150 per head for 20 persons or less is translated as
mathematical expression as:150𝑥 , 𝑖𝑓 0 ≤ 𝑥 ≤ 20
➢ P130 per head for 21 to 50 persons is translated as mathematical
expression as: 130𝑥 , 𝑖𝑓 21 ≤ 𝑥 ≤ 50
➢ P110 per head for 51 to 100 persons is translated as
mathematical expression as: 110𝑥 , 𝑖𝑓 51 ≤ 𝑥 ≤ 100
➢ For 100 or more persons, the cost is at P100 per head is
translated as mathematical expression as: 100𝑥 , 𝑖𝑓 x > 100
Step 4: Combine the function notation and subfunctions using equal (=) sign.
The result will be the piecewise function that represent the cost of renting (r)
in terms of number of days (x).
150x if 0 ≤ 𝑥 ≤ 20 , x ∈ ℕ
130x if 21 ≤ 𝑥 ≤ 50, x ∈ ℕ
110x if 51 ≤ 𝑥 ≤ 100, x ∈ ℕ
100x if x > 100, x ∈ ℕ
Note:
• 20 persons or less means 20 persons and below (less than)
• 100 or more persons means 100 persons and above (greater than)
• 51 to 100 persons means less than or equal to 100 but greater than or
equal to 51.
• More than 100 means greater than 100.
• x ∈ ℕ, 𝑟𝑒𝑎𝑑𝑠 𝑎𝑠 𝑥 𝑖𝑠 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠. X should be an
element of natural number because the number of attendees should be
in whole number. There will be never 1.5 persons/attendees.
f(x) =
Tips: The number of input
values or independent variable
gives the number of
subfunctions of the piecewise
function.

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I. Activity
A. Model it with Functions
After knowing more on functions, let us now apply these in the real-life
situations. You are task to model the situation with functions, by constructing
mathematical model to represents the real-life scenario. Put your answer on
a sheet of paper.
Scenario # 1:
A computer shop operates for a cause at this time of pandemic and
charges 25 pesos for the first two hours, but for the succeeding hours, an
additional charge of 15 pesos per hour is added.
1-2. The number of hours (h) is the _________and the charges in using the
computer (c) is _________.
1. The function notation to be used to represent the situation is ________.
2. Using the information given in the scenario, the subfunctions with
corresponding intervals are _________ and _________.
3. The piecewise function that represents the charges in using the
computer in terms of the number of hours is _________.
Scenario # 2:
You plan to bake and sell pandesal varieties each day to raise funds for
buying face masks for the frontliners facing the pandemic. You sell it for P8.00
each for the first 80 pandesals brought by a customer. After the first 80
pandesals up to 160 pandesals sold by the same customer you lower the price
to P5.50 each. After 160 pandesals sold by the same customer the price will
decrease to P3.00 each.
6-7. The input used in the situation is _________ and the output is
_________.
8. The function notation to be used to represent the situation is ________.
9. Using the information given in the scenario, the subfunctions with
corresponding intervals are _________ , _________ , and __________.
10. The piecewise function that describes the of pandesal is ________

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Key to Corrections
A. Model it with Functions
1. input 6. number of pandesal (n)
2. output 7. Sales (s) or price (p)
3. h(c) 8. s(n), may vary according to
4. 25 if 0 ≤ c ≤ 2 variables used
25 + 15(c – 2) if c > 2 9. 8n if 0 ≤ n ≤ 80
5. ℎ(𝑐) = {
25 0 ≤ 𝑐 ≤ 2
25 + 15(𝑐 − 2) 𝑐 > 2
5.50n if 81≤ n ≤ 160
3n if n > 160
10. s(n)= {
8n if 0 ≤ n ≤ 80
5.50n if 81 ≤ n ≤ 160
3n if n > 160
References
Smith, Jessica. 2019. “How Are Linear Equations Used in Everyday Life?”
Sciencing, N.p. Last modified August 7, 2020.
https://sciencing.com/linear-equations-used-everyday-life-
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Learning, N.p. Last accessed August 10,2020.
https://courses.lumenlearning.com/waymakercollegealgebra/chapter/
evaluate-and-solve-functions/.
N.p. n.d. “Function Notation (Examples, Solutions, Videos).”
Onlinemathlearning. Last accessed August 11,2020.
https://www.onlinemathlearning.com/function-notation.html.
"Function Application for the Real World." Study.com. September 15, 2017.
https://study.com/academy/lesson/function-application-for-the-real-
world.html.
"Functions: Identification, Notation & Practice Problems." Study.com.
January 15, 2013. https://study.com/academy/lesson/how-to-use-
function-notation.html.
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