GENERAL MATHEMATICS FOR GRADE 11- FUNCTIONS.pptx

At the end of the session, the teacher-participants are expected to:
1. represents real-life situations using functions, including piece-wise
functions.
2. evaluates a function.
3. performs addition, subtraction, multiplication, division, and composition
of functions.
4. solves problems involving functions.

PRIMING ACTIVITY
I’m going to show you a series of graphs.
Determine whether or not these graphs are
functions.
You do not need to draw the graphs in your
notes.

PRIMING ACTIVITY
The boat is sinking. Group yourselves by
What value will complete the table ?
5

PRIMING ACTIVITY
If f (x ) = k x 2
, and f ( 2 ) = 12, then k equals
3

PRIMING ACTIVITY
6

MANUEL S. ENVERGA UNIVERSITY FOUNDATION
An Autonomous University
BASIC EDUCATION DEPARTMENT
PRIMING ACTIVITY
10

ACTIVITY 1 (Group Work) (15 min)
1. You are given four graphs, four equations, four tables, and four rules.
2. Your task is to match each graph with an equation, a table and a rule.
3. Compare your results and describe their properties.
4. The team leaders of the groups report their conclusions to the whole
class.
Sorting Functions

Allow 1 point for each two correct answers ( 6 points).
Sorting Functions Rubric

Allow 1 point for correct explanation such as:
Equation C is a quadratic curve that passes through the origin and is
symmetrical about the y axis, so this is Graph A.
Equation D is the equation of a straight line, so this is Graph B.

Allow 1 point for correct explanation such as:
Equation B is a quadratic curve that passes through the origin and is
symmetrical about the x axis, so this is Graph C.
Equation A is an inverse (hyperbolic) function: the graph
approaches, but does not cross the axes (the axes are asymptotes) so
this is Graph D.

ANALYSIS
1. How did you find the activity matching each graph with an equation, a
table or a rule?
2. What did you find difficult about this task?
3. Have you encountered any problem or difficulty in performing your
task? Cite them.
4. What are some common errors which students may commit? How can
we prevent such error? Feel free to offer a suggestion.

ABSTRACTION
LEARNING AREA STANDARD:
At the end of the course, the students must know how to solve problems
involving rational, exponential and logarithmic functions; to solve business-
related problems; and to apply logic to real-life situations.
CONTENT STANDARD:
The learner demonstrates understanding of key concepts of functions.

ABSTRACTION
PERFORMANCE STANDARD:
The learner is able to accurately construct mathematical models to
represent real-life situations using functions.

ABSTRACTION
LEARNING COMPETENCIES:
The learners ...
a. represents real-life situations using functions, including piece-wise
functions.
b. evaluates a function.
c. performs addition, subtraction, multiplication, division, and composition
of functions.
d. solves problems involving functions.

RELATIONS AND FUNCTIONS
Is a relation a function?
According to the textbook, “a function is…a
relation in which every input is paired with
exactly one output”
What is a function?

RELATIONS AND FUNCTIONS
Focus on the x-coordinates, when given a relation.
If the set of ordered pairs have different x-coordinates,
it is a function.
Y-coordinates have no bearing in determining functions.
If the set of ordered pairs have same x-coordinates,
it is NOT a function.

FUNCTION AS A MACHINE
A function is a rule which operates on an input and produces a single
output from that input.

Write down the output from the function shown in the figure when the
input is a) 4 b) – 3 c) x d ) t

What is the domain? { 4, -5, 0, 9, -1 }
What is the range? { -2, 7 }
Input 4 –5 0 9 –1
–2 7
Output

FUNCTIONS AND RELATIONS AS A SET OF
ORDERED PAIRS
If a relation have the same first coordinate but different second
coordinates, then it is not a function

Is this a function?
Hint: Look only at the x-coordinates
YES
{( , ),( , ),( , ),( , ),( , ),( , )}
    
0 5 1 4 2 3 3 2 4 1 5 0
ORDERED PAIRS

Is this a function?
Hint: Look only at the x-coordinates
NO
{(– , ),( , ),( , ),( , ),( , ),(– , )}
   
1 7 1 0 2 3 0 8 0 5 2 1
ORDERED PAIRS

Which mapping represents a function?
Choice 1
Choice One
3
1
0
–1
2
3
Choice Two
2
–1
3
2
3
–2
0
ORDERED PAIRS

Which mapping represents a function?
B
A B
ORDERED PAIRS

What is the domain?
{( , ),( , ),( , ),( , ),( , ),( , )}
    
0 5 1 4 2 3 3 2 4 1 5 0
{ 0, 1, 2, 3, 4, 5 }
What is the range? { -5, -4, -3, -2, -1, 0 }
ORDERED PAIRS

Find the Domain and Range of a Relation.
a. {(–5, 7), (3, 5), (6, 7), (12, –4)}
Domain: {–4,-2, 2,3, 4};
Range: {-5,–2, 0, 3, 4}
b.
Domain: {–5, 3, 6, 12};
Range: {–4, 5, 7}
ORDERED PAIRS

Determine if Relations are Functions
a. {(–3, 6), (2, 5), (0, 6), (17, –9)}
b. {(4, 5), (7, –3), (4, 10), (–6, 1)}
c. {(–2, 3), (0, 3), (4, 3), (6, 3), (8, 3)}
d. I y – 5 I = x + 3
e. y = x2
− 3 x + 2
f. 4 x – 8 y = 24
Function
Not a function Four has two values
Function
Not a function One has two values
Function
Function
ORDERED PAIRS

FUNCTIONS AND RELATION AS
AS A TABLE OFVALUES
We can present the ordered pairs in a list or table.

FUNCTIONS AS A GRAPH IN THE
CARTESIAN PLANE
We can use a graph to indicate the ordered pairs. The graph can show
distinct ordered pairs, or it can show all the ordered pairs on a line or
curve.

c.
Domain is { x | – 5 ≤ x ≤ 5 }
in set-builder notation
or [ – 5, 5 ] in interval notation
Range is { y | – 3 ≤ y ≤ 4}
or [ – 3, 4 ] in interval notation
CARTESIAN PLANE

d.
Domain is { x | x ≤ 3 }
or (– ∞, 3] in interval notation
Range is in set-builder notation
ℝ
or ( – ∞, ∞ ) in interval notation
Domain is or ( – ∞, ∞ )
ℝ
Range is { y | y ≥ 0 } or ( 0, ∞ )
CARTESIAN PLANE

Summary of Graphs of Common Functions
f(x) = c
y = x
x
y 
x
y  y = x2
y = x 3

VERTICAL LINE TEST
Vertical Line Test
If a vertical line intersects (crosses or touches) the graph of a
relation at more than one point, then the relation is not a
function.
If every vertical line intersects the graph of a relation at no more
than one point, then the relation is a function

What is a function?
FUNCTION FUNCTION
NOT A FUNCTION
FUNCTION
FUNCTION
NOT A FUNCTION
NOT A FUNCTION

What is a function?
1 2
3
5
7
4
6

Functions in the real world
Money as a function of time. You never have more than one amount of
money at any time because you can always add everything to give one
total amount.
Temperature as a function of various factors. Temperature is a very
complicated function because it has so many inputs, including: the time of
day, the season, the amount of clouds in the sky, the strength of the wind,
where you are and many more.
Location as a function of time. You can never be in two places at the same
time.
FUNCTIONS AS REPRESENTATIONS
OF REAL LIFE SITUATIONS

Which situation represents a function?
a
a. The items in a store to their prices on a certain date
Solution:
b. Types of fruits to their colors

Real-life situation
Identify whether the relation that exists between each
of the following pairs indicates a functions or not.
1. A jeepney to its plate number.
2. A teacher to his/her cellular phone.
3. A pen to its ink color.
4. A student to his/her student number.
F
Not F
Not F
F

The number of shoes in x pairs of shoes can be expressed by the
equation y = 2x.
Whole numbers
What subset of the real numbers makes sense for the domain?
What would make sense for the range of the function?
Zero and the even numbers

The # of pairs of shoes
What is the independent variable?
What is the dependent variable?
The total # of shoes.
The number of shoes in x pairs of shoes can be expressed by the
equation y = 2x.

Mr. Landry is driving to his hometown. It takes four hours to get there. The
distance he travels at any time, t, is represented by the function d = 55 t
(his average speed is 55 mph).
Write an inequality that represents the domain in real life.
Write an inequality that represents the range in real life.
0 4
x
 
0 220
y
 

The time that he drives.
The total distance traveled.
Mr. Landry is driving to his hometown. It takes four hours to get there. The
distance he travels at any time, t, is represented by the function d = 55 t
(his average speed is 55 mph).

Johnny bought at most 10 tickets to a concert for him and his friends. The
cost of each ticket was $12.50. Complete the table below to list the possible
domain and range.
What is the independent variable? What is the dependent variable?
The number of tickets bought. The total cost of the tickets.
1 2 3
12.50 25.00 37.50
4
50
5
62.50
6 7 8 9 10
75 125
112.50
100
87.50

Pete’s Pizza Parlor charges $5 for a large pizza with no toppings. They
charge an additional $1.50 for each of their 5 specialty toppings (tax is
included in the price).
Jorge went to pick up his order. They said his total bill was $9.50. Could
this be correct? Why or why not?
Yes
One pizza with 3 toppings cost $9.50

Susan went to pick up her order. They said she owed $10.25. Could this
be correct? Why or why not?
No
One pizza with 4 toppings cost $11

The number of toppings
The cost of the pizza

PIECEWISE FUNCTIONS
Up to now, we’ve been looking at functions represented by a
single equation.
In real life, however, functions are represented by a
combination of equations, each corresponding to a part of the
domain.
These are called piecewise functions.

PIECEWISE FUNCTIONS
 








1
,
1
3
1
,
1
2
x
if
x
x
if
x
x
f
•One equation gives the value of f(x) when x ≤ 1
•And the other when x>1

Evaluate f(x) when x=0, x=2, x=4








2
,
1
2
2
,
2
)
(
x
if
x
x
if
x
x
f
•First you have to figure out which equation to use
•You NEVER use both
X=0
This one fits
Into the top
equation
So:
0+2=2
f(0)=2
X=2
This one fits here
So:
2(2) + 1 = 5
f(2) = 5
X=4
This one fits here
So:
2(4) + 1 = 9
f(4) = 9

Graph:









1
,
3
1
,
)
( 2
3
2
1
x
if
x
x
if
x
x
f
•For all x’s < 1, use the top graph (to the left of 1)
•For all x’s ≥ 1, use the bottom graph (to the
•right of 1)

1
2
3
, 1
2
( )
3, 1
x if x
f x
x if x

 


  


x=1 is the breaking
point of the graph.
To the left is the top
equation.
To the right is the
bottom equation.

Graph:
1, 2
( )
1, 2
x if x
f x
x if x
  

  

Point of Discontinuity

Step Functions
















4
3
,
4
3
2
,
3
2
1
,
2
1
0
,
1
)
(
x
if
x
if
x
if
x
if
x
f

















4
3
,
4
3
2
,
3
2
1
,
2
1
0
,
1
)
(
x
if
x
if
x
if
x
if
x
f

Graph :























0
1
,
4
1
2
,
3
2
3
,
2
3
4
,
1
)
(
x
if
x
if
x
if
x
if
x
f

Special Step Functions
Two particular kinds of step functions are called ceiling functions
( f (x)= and floor functions ( f (x)= ).
In a ceiling function, all nonintegers are rounded up to the nearest
integer.
An example of a ceiling function is when a phone service
company charges by the number of minutes used and always
rounds up to the nearest integer of minutes.
x
 
  x
 
 

Special Step Functions
In a floor function, all nonintegers are rounded down to the
nearest integer.
The way we usually count our age is an example of a floor
function since we round our age down to the nearest year and do
not add a year to our age until we have passed our birthday.
The floor function is the same thing as the greatest integer
function which can be written as f (x)=[x].

Greatest integer function: f(x) = [x]
For example:
[2.1] = 2, i.e. greatest integer less than or equal to 2.1 is 2, similarly
[–2.1] = –3
[2] = 2
[3 . 9] = 3
[–3 . 9] = –4
PIECEWISE FUNCTIONS

PIECEWISE FUNCTIONS
A store charges Php150 per t-shirt for orders of 50 or fewer, Php 135 per
t-shirt for orders of 75 or fewer but more than 50 t-shirts, and Php 125 per
t-shirt for orders of more than 75 t-shirts. Which function best represents
the printing cost C of x number of t-shirts?
(a)
(b) Answer: C
(c)











x
76
if
,
125
5
7
x
51
if
,
135
50
x
0
if
,
150
)
(
x
x
x
x
C











x
76
if
,
125
5
7
x
51
if
,
135
50
x
0
if
,
150
)
(
x
x
x
x
C











x
76
if
,
125
5
7
x
51
if
,
135
50
x
0
if
,
150
)
(
x
x
x
x
C

FUNCTIONS
Symmetric about the y axis
Symmetric about the origin

2
-7 -6 -5 -4 -3 -2 -1 1 5 7
3
0 4 6 8
7
1
2
3
4
5
6
8
-2
-3
-4
-5
-6
-7
For an even function: for every point (x, y) on the
graph, the point (-x, y) is also on the graph.
Even functions have y-axis Symmetry

2
-7 -6 -5 -4 -3 -2 -1 1 5 7
3
0 4 6 8
7
1
2
3
4
5
6
8
-2
-3
-4
-5
-6
-7
For an odd function: for every point (x, y) on the
graph, the point (-x, -y) is also on the graph.
Odd functions have origin Symmetry

2
-7 -6 -5 -4 -3 -2 -1 1 5 7
3
0 4 6 8
7
1
2
3
4
5
6
8
-2
-3
-4
-5
-6
-7
We wouldn’t talk about a function with x-axis symmetry
because it wouldn’t BE a function.
x-axis Symmetry

Even, Odd or Neither?
Graphically

f x x x
( )  
3
Graphically



Graphically

 
Graphically

Given
A function is even if for
every number x in the domain.
If you plug a –x into the function and you get the
original function back again, the function is even.
Same!
Even Function

Given
A function is odd if for
If you plug a –x into the function and you get the negative of
the function back (all terms change signs), the function is odd.
Odd Function
ALL signs of the
terms changed!

A function is even if for
If you plug a –x into the function and you get the
original function back again, the function is even.
  1
2
5 2
4


 x
x
x
f Is this function even?
  1
2
5
1
)
(
2
)
(
5 2
4
2
4








 x
x
x
x
x
f
YES
  x
x
x
f 
 3
2 Is this function even?
  x
x
x
x
x
f 






 3
3
2
)
(
)
(
2
NO

A function is odd if for
If you plug a –x into the function and you get the negative of
the function back (all terms change signs), the function is odd.
  1
2
5 2
4


 x
x
x
f Is this function odd?
  1
2
5
1
)
(
2
)
(
5 2
4
2
4








 x
x
x
x
x
f
NO
  x
x
x
f 
 3
2 Is this function odd?
  x
x
x
x
x
f 






 3
3
2
)
(
)
(
2
YES

Odd, Even, or Neither?
Even Function

Neither Odd or Even

Odd Function

EVALUATING FUNCTIONS
f(x) means function of x and is read “f of x.”
f(x) = 2x + 1 is written in function notation.
The notation f(1) means to replace x with 1 resulting in the function
value.
f(1) = 2x + 1
f(1) = 2(1) + 1
f(1) = 3

Function Notation
Given g(x) = x 2
– 3, find g(-2) .
g(-2) = x 2
– 3
g(-2) = (-2) 2
– 3
g(-2) = 1

Function Notation
Given f(x) = 2x2
– 3x , find the following.
a. f(3) b. 3f(x) c. f(3x)
f(3) = 2x2
– 3x
f(3) = 2(3)2
– 3(3)
f(3) = 2(9) - 9
f(3) = 9
3f(x) = 3(2x2
– 3x)
3f(x) = 6x2
– 9x
f(3x) = 2x2
– 3x
f(3x) = 2(3x)2
– 3(3x)
f(3x) = 2(9x2
) – 3(3x)
f(3x) = 18x2
– 9x

For each function, evaluate f(0), f(1.5), f(-4),
f(0) =
f(1.5) =
f(-4) =
3
4
4

f(0) =
f(1.5) =
f(-4) =
1
3
1

f(0) =
f(1.5) =
f(-4) =
– 5
1
1

Given f(x) = x2
– 4x + 7 Find.
h
x
f
h
x
f )
(
)
( 

h
x
x
h
x
h
x ]
7
4
[
]
7
)
(
4
)
[( 2
2








h
x
x
h
x
h
xh
x 7
4
7
4
4
2 2
2
2









h
h
h
xh 4
2 2



h
h
x
h )
4
2
( 

 = 2x + h - 4

OPERATIONS ON FUNCTIONS

USING OPERATIONS ON FUNCTIONS

COMPOSITION OF FUNCTIONS and DOMAIN

EVALUATING COMPOSITE FUNCTIONS

APPLICATION
1. A ball is thrown straight up, from 3 m above the ground, with a velocity
of 14 m/s. When does it hit the ground?
2. A 3 hour river cruise goes 15 km upstream and then back again. The
river has a current of 2 km an hour. What is the boat's speed and how long
was the upstream journey?
3. Your company is going to make frames as part of a new product they
are launching. The frame will be cut out of a piece of steel, and to keep
the weight down, the final area should be 28 cm2
.The inside of the frame
has to be 11 cm by 6 cm. What should the width x of the metal be?

APPLICATION
1. h = 3 + 14t − 5 t 2
2. total time = 15/(x−2) + 15/(x+2) = 3 hours
3. x = 0.8 cm (approx.)
The ball hits the ground after 3 seconds!
Boat's Speed = 10.39 km/h
upstream journey = 15 / (10.39−2) = 1.79 hours = 1 hour 47min
downstream journey = 15 / (10.39+2) = 1.21 hours = 1 hour 13min

TEACH is
TOUCH LIVES
EVER to an Individual
Learner.
Closure

Group 1 : Module 1
Group 2 : Module 2
Group 3 : Module 3
Group 4 : Module 4
Group 5 : Module 5

GENERAL MATHEMATICS FOR GRADE 11- FUNCTIONS.pptx

More Related Content

What's hot

Similar to GENERAL MATHEMATICS FOR GRADE 11- FUNCTIONS.pptx

Recently uploaded

GENERAL MATHEMATICS FOR GRADE 11- FUNCTIONS.pptx

Editor's Notes