Functions, Introduction This is the slide presentation I did for BSIT and ACT students

1. C A L C UL US 4 B S I T - A C T
FUNCTIONS

2. CALCULUS
FUNCTIONS
Functions – is a rule of correspondence between two
sets X and Y such that
each element in the first set X is paired to one and
only element in the second set Y
X
Y
x
y
f
Figure 1.1 Function as a Rule
A
B
BA
to

3. Key words: pairing and only one element y in Y is
paired with each x in X.
a
b
m
N
o
Not a function
a
b
m
N
o
a function
CALCULUS
FUNCTIONS

4. Functions produce a set of ordered pairs(x,y) .
2
3
5
4
9
25
f
F={(2,4), (3,9), (5, 25)}
CALCULUS
FUNCTIONS

5. Fundamental Theorem of Analytic Geometry
Each ordered pair of real numbers correspond to a
point (x,y) on the Cartesian plane.
Graph – the collection of points on the Cartesian
plane (graph of the function
CALCULUS
FUNCTIONS

6. • F={(2,4), (3,9), (5, 25)}
CALCULUS
FUNCTIONS

7. Functions in real life situations
A. A student earns a general-weighted average
upon graduation in LPU.
B. Each item in a grocery has a price.
C. Each government employee has corresponding
asset and liability as stated in the government
SALN.
CALCULUS
FUNCTIONS

8. Functions in real life
(x,y)
𝑦 = 𝑓 𝑥
Domain – is the set of all admissible values of x.
Range – is the set of all resulting values of y
CALCULUS
FUNCTIONS

9. Identify the domain and range for the following
functions.
A. A student earns a general-weighted average
upon graduation in LPU.
B. Each item in a grocery has a price.
C. Each government employee has
corresponding asset and liability as stated in
the government SALN.
D. Monthly sales report of a Camella Homes
CALCULUS
FUNCTIONS

10. Given a function 𝑦 = 𝑓 𝑥
Independent variable – X
Dependent variable – Y
Identify the dependent and independent variable in
the following
A. A student earns a general-weighted average
upon graduation in LPU.
B. Each item in a grocery has a price.
C. Each government employee has
corresponding asset and liability as stated in
the government SALN.
D. Monthly sales report of a Camella Homes
CALCULUS
FUNCTIONS

11. More on graphs. Not all graphs of functions in real life
are continuous curves, especially in business.
Monthly
sales in
hundred
thousands
Months (1-Jan, 2-Feb,. . .et.
CALCULUS
FUNCTIONS

12. Why would distinct points be connected by a curve?
a. To show trending.
b. To effectively show the behavior of the function
c. To fill-in missing data with educated guesses.
CALCULUS
FUNCTIONS

13. Functions defined by equations
In mathematics and business related situations,
functions are described using equations.
𝑓 𝑥 = 𝐶𝑥 + 𝐷
where C is the cost per unit of production and D is
the fixed cost. (Cost function)
𝑔 𝑡 = 𝐼(1.08) 𝑡
where I is the initial investment and t, number of
years. (Accumulated value function)
CALCULUS
FUNCTIONS

14. 𝑅 𝑥 = 𝑝𝑥 − 𝑐𝑥 = 𝑝 − 𝑐 𝑥
Profit function, p is the price per unit of merchandize
sold, and c is the cost per unit.
CALCULUS
FUNCTIONS

15. Input-the value of the independent variable
Output – the value of the dependent variable
CALCULUS
FUNCTIONS

16. CALCULUS FOR BUSINESS
FUNCTION
Types of functions
a. Linear function
𝑦 = 𝑎𝑥 + 𝑏

17. b. Quadratic function
𝑦 = 𝑎𝑥2 − 𝑏𝑥 + 𝑐
CALCULUS
FUNCTIONS

18. c. Exponential function
𝑦 = 𝐶𝑏 𝑥, 𝐶 initial value, 𝑥 ∈ 𝑅
𝑦 = 2𝑒2
CALCULUS
FUNCTIONS

19. 𝑦 = 2𝑒−𝑥
CALCULUS
FUNCTIONS

20. CALCULUS FOR BUSINESS
FUNCTIONS
c. Log function (Logistic function)
𝑦 = 𝑙𝑜𝑔 𝑏 𝑥, 𝑏 ∈ 𝑅+, 𝑥 ∈ 𝑅
𝑦 = ln 𝑥, base e
𝑦 = log 𝑥, base 10

21. 𝑦 = ln 𝑥
CALCULUS
FUNCTIONS

22. d. Polynomial function
𝑦 = 𝑎𝑥 𝑛
+ 𝑏𝑥 𝑛−1
+ ⋯ 𝑐
𝑥 ∈ 𝑅,
𝑎, 𝑏, … 𝑐, ∈ 𝑅 coefficients
𝑛 ∈ 𝑁
CALCULUS
FUNCTIONS

23. • 𝑦 = 5 − 𝑥 + 2𝑥2 − 4𝑥3
CALCULUS
FUNCTIONS

24. CALCULUS
FUNCTIONS
Conditional function (piece-wise functions)
 
 
 





condition3if
2conditionif
1conditionif
xh
xg
xf
y

25. 








x
xx
x
y
8if0
84if5.0
40if4
CALCULUS
FUNCTIONS

26. Conditional function
CALCULUS
FUNCTIONS

27. CALCULUS
FUNCTIONS

28. CALCULUS FOR BUSINESS
FUNCTIONS