1. Week 1_ Functions and Evaluate Functions.pptx

CHAPTER 1
KEY CONCEPTS OF
FUNCTIONS

1.What do you call a relation where each element in the
domain is related to only one value in the range by some
rules?
a. Function b. Range c. Domain d. Independent
2. Which of the following relations is/are function/s?
a. x = {(1,2), (3,4), (1,7), (5,1)}
b. g = {(3,2), (2,1), (8,2), (5,7)}
c. h = {(4,1), (2,3), (2, 6), (7, 2)}
d. y = {(2,9), (3,4), (9,2), (2,7)}

3. In a relation, what do you call the set of x values or the
input?
a.Piecewise b. Range c. Domain d. Dependent
4. What is the range of the function shown by the diagram?
a. R:{3, 2, 1}
b. R:{a, b}
c. R:{3, 2, 1, a, b}
d. R:{all real numbers}

Example 1: Which of the following relations are
functions?
f= {(1,2), (2,2), (3,5), (4,5)}
g= {(1,3), (1,4), (2,5), (2,6), (3,7)}
h= {(1,3) (2,6), (3,9), … , (n, 3n)}

Example 2: Which of the following relations are
functions?

Practice:
Which of the following relations are functions?
R= {(9,10, (-5, -2), (2, -1), (3, -9)}
S= {(6, a), (8, f), (6, b), (-2, p)}
T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)}
V= {(6, a), (8, f), (7, a), (-2, p)}

Functions can be represented in different ways.

Domain: a set of first elements in a relation (all of the x values).
These are also called the independent variable.
Range: The second elements in a relation (all of the y values).
These are also called the dependent variable.
Table of Values

Function as Mapping

Identify the Domain and Range. Then tell if the relation is a
function.
Input Output
-3 3
1 1
3 -2
4
Domain = {-3, 1,3,4}
Range = {3,1,-2}
Function?
Yes: each input is mapped
onto exactly one output

Input Output
-3 3
1 -2
4 1
4
Identify the Domain and Range. Then tell if the
relation is a function.
Domain = {-3, 1,4}
Range = {3,-2,1,4}
Function?
No: input 1 is mapped onto
Both -2 & 1
Notice the set notation!!!

1. {(2,5) , (3,8) , (4,6) , (7, 20)}
2. {(1,4) , (1,5) , (2,3) , (9, 28)}
3. {(1,0) , (4,0) , (9,0) , (21, 0)}

Example
07/30/2025 02:22 PM 1-6 RELATIONS AND FUNCTIONS 24
•Is this a function?
•Hint: Look only at the x-coordinates
NO
{(-1,-7) , (1,0) , (2,-3) , (0, -8) (0, 5 ), (-2, -1)}

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

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

Example
A. B.
Which mapping represents a function?
B

Functions as a graph in the Cartesian plane

Cartesian Coordinate System
Cartesian coordinate plane
x-axis
y-axis
origin
quadrants
Functions as a graph in the Cartesian plane

A Relation can be represented by a set of ordered pairs of
the form (x,y)
Quadrant I
X>0, y>0
Quadrant II
X<0, y>0
Quadrant III
X<0, y<0
Quadrant IV
X>0, y<0
Origin (0,0)

Plot: (-3,5) (-4,-2) (4,3) (3,-4)

Ex4 . Graph y = 3x – 1.
x 3x-1 y

Ex 5. Graph y = x² - 5
x x² - 5 y
-3
-2
-1
0
1
2
3

The Vertical Line Test
A graph represents a function if and only if
each vertical line intersects the graph at
most once.

(-3,3)
(4,4)
(1,1)
(1,-2)
Use the vertical line test to visually check if the relation is a
function.
Function?
No, Two points are on
The same vertical line.

(-3,3)
(4,-2)
(1,1)
(3,1)
Use the vertical line test to visually check if the relation is a
function.
Function?
Yes, no two points are
on the same vertical line

Examples
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.

Which of the following equation represents a
function?

Function as a Machine

INPUT
(DOMAIN)
OUTPUT (RANGE)
FUNCTION
MACHINE
In order for a relationship to be a function…
EVERY INPUT MUST HAVE AN OUTPUT
TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT
Functions
ONE INPUT CAN HAVE ONLY ONE OUTPUT

Definition:
Piecewise Function –a function defined by
two or more functions over a specified
domain.
The rule for a piecewise function is different
for different parts, or pieces, of the domain.
For instance, movie ticket prices are often
different for different age groups. So the
function for movie ticket prices would assign a
different value (ticket price) for each domain
interval (age group).

When using interval notation, square brackets [ ]
indicate an included endpoint, and parentheses
( ) indicate an excluded endpoint.
Remember!

What do they look like?
f(x) =
x2
+ 1 , x  0
x – 1 , x  0
You can EVALUATE piecewise
functions.
You can GRAPH piecewise functions.

Evaluating Piecewise Functions:
Evaluating piecewise functions is just
like evaluating functions that you are
already familiar with.
f(x) =
x2
+ 1 , x  0
x – 1 , x  0
Let’s calculate f(2).
You are being asked to find y when
x = 2. Since 2 is  0, you will only
substitute into the second part of the
function.
f(2) = 2 – 1 = 1

f(x) =
x2
+ 1 , x  0
x – 1 , x  0
Let’s calculate f(-2).
You are being asked to find y when
x = -2. Since -2 is  0, you will only
substitute into the first part of the
function.
f(-2) = (-2)2
+ 1 = 5

Your turn:
f(x) =
2x + 1, x  0
2x + 2, x  0
Evaluate the following:
f(-2) = -3
?
f(0) = 2
?
f(5) = 12
?
f(1) = 4?

Your turn:
f(x) =
2x + 1, x  0
2x + 2, x  0
f(-2) = -3
-3
f(0) = 2
2
f(5) = 12
12
f(1) = 44

One more:
f(x) =
3x - 2, x  -2
-x , -2  x  1
x2
– 7x, x  1
f(-2) = 2?
f(-4) = -14?
f(3) = -12
?
f(1) = -6
?

One more:
f(x) =
3x - 2, x  -2
-x , -2  x  1
x2
– 7x, x  1
f(-2) = 22
f(-4) = -14
-14
f(3) = -12
-12
f(1) = -6
-6

Graphing Piecewise Functions:
f(x) =
x2
+ 1 , x  0
x – 1 , x  0
Determine the shapes of the graphs.
Parabola and Line
Determine the boundaries of each graph.
Graph the parabola
where x is less than
zero. 





Graph the line where x
is greater than or
equal to zero.














3x + 2, x  -2
-x , -2  x  1
x2
– 2, x  1
f(x) =
Graphing Piecewise Functions:
Determine the shapes of the graphs.
Line, Line, Parabola
Determine the boundaries of each graph.










 


When using interval notation, square brackets [ ]
indicate an included endpoint, and parentheses
( ) indicate an excluded endpoint.
Remember!
Use piecewise functions to describe
real-world situations.

Example: A user is charged 300 monthly for a particular mobile
₱
plan, which includes 100 free text messages. Messages in excess of
100 are charged 1 each. Represent the amount a consumer pays
₱
each month as a function of the number of messages m sent in a
month.
Solution: Let t (m) represent the amount paid by the consumer each
month. It can be expressed by the piecewise function

Example:A jeepney ride costs 8.00 for the first 4 kilometers, and
₱
each additional kilometer adds 1.50 to the fare. Use a piecewise
₱
function to represent the jeepney fare in terms of the distance d in
kilometers.
Solution: The input value is the distance and the output is the cost
of the jeepney fare. If F(d) represents the fare as a function of
distance, the function can be represented as follows:

Example:A jeepney ride costs 8.00 for the first 4 kilometers, and
₱
each additional kilometer adds 1.50 to the fare. Use a piecewise
₱
function to represent the jeepney fare in terms of the distance d in
kilometers.
Solution: The input value is the distance and the output is the cost
of the jeepney fare. If F(d) represents the fare as a function of
distance, the function can be represented as follows:
F(d) = 8 if 0 < d ≤ 4
8 + 1.5(d-4) if d > 4

3. Which of the following letters will pass the
vertical line test?
A. V
B. X
C. Y
D. Z

4. To evaluate a function is to
______________ the variable in the
function with a value from the function’s
domain and compute the result.
A. omit
B. simplify
C. skip
D. substitute

5. The altitude of a plane is a function of the
times since take off. What is the dependent
variable?
A. Time
B. Speed
C. Altitude
D. All of the above

8. If ( ) = −8 + 1, find (−2).
𝑔 𝑥 𝑥 𝑔
A.-17
B. 8
C. 10
D. 17

6. If ( ) = + 2 – 1, find (−1).
ℎ 𝑥 𝑥 ℎ
A.-5
B.-3
C. 0
D. 3

7. If , find (0).
𝑓
A.-5
B.-3
C. 0
D. 3

9. If ( ) = 8 + 1, find (2).
𝑔 𝑥 𝑥 𝑔
A.-7
B. 1
C. 10
D. 17

10. If ( ) = + 1, find (-2).
𝑔 𝑥 𝑥 𝑔
A.-1
B. 3
C.-2
D. 17

1. If ( ) = 8 + 1, find (2).
𝑔 𝑥 𝑥 𝑔
A.-7
B. 1
C. 10
D. 17

2. If ( ) = + 2 – 1, find (0).
ℎ 𝑥 𝑥 ℎ
A. 1
B.-1
C. 5
D.-5

3. If ( ) = + 1, find (-2).
𝑔 𝑥 𝑥 𝑔
A.-1
B. 3
C.-2
D. 17

4. A person can encode 1000 words in every hour
of typing job. Which of the following expresses the
total words W as a function of the number n of
hours that the person can encode?
a. ( ) = 1000+
𝑊 𝑛 𝑛
b. ( ) =
𝑊 𝑛
c. ( ) = 1000
𝑊 𝑛 𝑛
d. ( ) = 1000 −
𝑊 𝑛 𝑛

INPUT FUNCTION OUTPUT
g(1)
g(x) = 2x -3
g(-3)
g(0)
5.

B. Find the output values of the functions when
the input value (x) is given.
INPUT FUNCTION OUTPUT
f(1)
f(x) = 2x -1
f(3)
f(0)
f(-2)
f(-1)

1. Week 1_ Functions and Evaluate Functions.pptx

1. Week 1_ Functions and Evaluate Functions.pptx

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