Group 2

Function and Application

Mathematics in the Modern World

GE 4

2021
Mathematics in the Modern World
GE 4

Group 2
Function and Application

Submitted by
Evangelista, Joel R.
Isiang, James G.
Marcillano, Jeremy
Requiestas, Jian Maru A.

Submitted to
Prof. Jemar Mejares

2021
Function and Application
One of the most complicated parts in algebra is the study of graphs and functions. The notion of a function
is a correspondence between 2 sets. The price of a stock can be determined by looking at the daily quote in the
newspaper; height of a building can be determined by dropping a small object and measuring the time it tasks
to hit the bottom; and the surface area of a circular disk can be determined if you know it’s radius. All of these
are everyday examples of functions.

A relation is any set of ordered pairs (x, y) of real numbers, for a function, we consider a correspondence or
a mapping, between 2 set x of y so that x is a member of X and y is a member of Y.

Definitions

A function f is a mapping that assigns to each element x of X a unique element of y of Y. the element y is
called the image of x under f and is denoted by f(x). The set x is called the domain of the function. The set of all
image of elements y is called the range of the function.

Illustration: A function as mapping different elements in the domain with different element in its image.

A function as mapping

Example 1: Different elements in the domain with different element in its image.

X= {a, b, c, d} f (a)=1

Domain is {a, b, c, d} f (b)=2

 Y={1, 2, 3, 4} f (c)=3

Range is {1, 2, 3, 4} f (d)=4

Example 2: Different elements in the domain have the same imagine in the range.

D={-2, -1, 2, 4, 5} R={3, -3, 0}

X maps into Y

f(-2)=3 f(2)=-3

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

Definition

If maps X into Y so that for any distinct elements x1 and x2 of X, f(x1) ≠ f(x2), then f is a one-to-one function
of X into Y.

Example 3: It illustrate a mapping that is not function.

D={-3, 3, 0} R={-2, -1, 2, 3, 4, 5}

This is not a function because -3 and 0 is associated with more than one image.
Definition

A function is a set of ordered pairs for which member x of the domain is associated with exactly one member f
(x) of the range.

Example 4: f ={(1, 2),(3, 5),(6, 9),(7, 11)}

f (1)=2

f (3)=5

f (6)=9

f (7)=11

Example 5: h = {(0, 0),(2, -2),(4, 6),(-4, 6)}

D = {0, 2, 4} and R = {0, -2, 2, 6, -6}

h(0) = 0

g(2) = 2

g(4) = {6

{-6

This is not a function; both 2 and 4 are associated with more

the one second component.

Functional Notation
The function is denoted by f, f(x) is the number associated by x.

f(x) = x + 1 or g(x) = x2 + 2x + 2

To emphasize the difference between f and f(x). f : x → x + 1

To define function, but simply uses the notation: f(x) = x + 1
To mean the set of all ordered pair ( x , y ), such that y = x + 1.
Example 1: Given f and g defined by f(x) = x + 1 and g(x) = x2 + 2x + 2, find the indicated values: (a) f(1), (b) g(2).

Solutions:

(a) f(1): f(x) = x + 1

f(1) = 1 + 1
f(1) = 2

(b) g(2): g(x) = x2 + 2x + 2

g(2) = 22 + 2(2) + 2 = 4 + 4 + 2
g(2) = 10

Example 2: Let F and G be defined by F(x) = x2 + 2 and G(x) = ( x + 3 )2. Find: (a) F( w + 3 ), (b) G( x – 2).

Solutions:

(a) F( w + 3 ): F(x) = x2 + 2
F( w + 3 ) = ( w + 3 )2 + 2 = ( w2 + 6w + 9 ) + 2
F( w + 3 ) = w2 + 6w + 11

(b) G( x – 2 ): G(x) = ( x + 3 )2
G( x – 2 ) = [( x – 2 ) + 3 ]2 = ( x + 1 )2
G( x – 2 ) = x2 + 2x + 1

Domain

The domain of an equation is the set of all x’s that we can plug into the equation and get back a real number
for y.

Examples:

(a) f(x) = 3x – 2
( 2𝑥−1 )(𝑥+3)
(b) g(x) =
𝑥+3
(c) h(x) = √𝑥 + 2

Solutions:

(a) All real numbers: ( −∞, ∞)

(b) All real numbers, x ≠ -3
(2𝑥−1)(𝑥+3)
Hence, 𝑔(𝑥) = = 2𝑥 − 1
(𝑥+3)
Therefore, the domain are all real numbers, x ≠ -3
(c) h has a meaning if x + 2 is nonnegative, that is
𝑥+2 ≥0 𝑜𝑟 𝑥 ≥ −2
The domain can be discarded as [−2, ∞).
Graphing Linear Function
Definition
A function f is a linear function if.

f(x) = mx + b

Where m and b are real numbers

Suppose that if m = 0, then f(x) = b, is called a constant function. if the domain of a constant function is a set
of real numbers, the graph of f(x) = b is a horizontal line.

If we let P1 (x1, y1) and P2 (x2, y2) be any point on a line and suppose x1 = x2, the line is parallel to the y-axis is
called a vertical line.

Definition

Let P1 (x1, y1) and P2(x2, y2) be distinct points on the line such that x1 ≠ x2, then

𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒄𝒉𝒂𝒏𝒈𝒆 𝒚𝟐 − 𝒚𝟏
𝒔𝒍𝒐𝒑𝒆 = =
𝒉𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝒄𝒉𝒂𝒏𝒈𝒆 𝒙𝟐 − 𝒙𝟏

The numerator y2 – y1 is often called the rise and the denominator x2 – x1, the run. From P1 to P2, if we use the
functional notation for the points P1( x1, f( x1 )) and P2 ( x2, f( x2)), then the slope is found by

𝒇(𝒙𝟐 ) − 𝒇(𝒙𝟏 )
𝒔𝒍𝒐𝒑𝒆 =
𝒙𝟐 − 𝒙 𝟏

Example 1: Draw the line passing through the points whose coordinates are given.
Then find the slope of each line.
 a. ( 2, -2 ) and ( -1, 3 )
b. ( -4, -1 ) and ( 1, 3 )

Solution:

3−(−2) 5
a. 𝑚 = = −
−1−2 3

3−(−1) 4
b. 𝑚 = =
1−(−4) 5

1
Example 2: Graph 𝑦 = 𝑥+2
2

1
Solution: By inspection, the y-intercept is 2 and the slope is 2. The line is graph by first plotting the y-intercept
( 0, 2 ), then finding a second point by counting out the slope, up 1 and over 2.

Example 3: Graph 2x + 3y – 6 = 0

Solution: Solve for y.

3y = -2x + 6
2
y = −3𝑥 + 2

2
The y-intercept is 2 and the slope is − 3.
 The graph is shown below.

Application of Linear Functions

Linear relationships are often used to model real-life situations. In order to create an equation and graph
to model the real-life situation, you need at least two data values related to the real-life situation. When the
data values have been represented graphically and the equation of the line has been determined, questions
relating to the real-life situation can be presented and answered.

When equations and graphs are used to model real-life situations, the domain of the graph is
sometimes xϵN. However, it is often more convenient to sketch the graph as though xϵR instead of showing the
function as a series of points in the plane.

Example:

A cab company charges $2.00 for the first 0.6 miles and $0.50 for each additional 0.2 miles.
This situation cannot be modeled with just one equation.
To graph this situation, the x-axis is the distance in miles and the y-axis is the cost in dollars. The first line
from A to B extends horizontally across the distance from 0 to 0.6 miles. The cost is constant at $2.00. The
equation for this constant function is y=2.00 or c=2.00. The second line from B to C and upward is not constant.

The equation that models the second graph can be determined by using the data points (0.6, 2.00) and (1,
3.00)

𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
 3− 2
𝑚= , Use the data points to calculate the slope.
1− 0.6
1
𝑚=
0.4

𝒎 = 𝟐. 𝟓

Graphing Quadratic Functions

The simplest quadratic function is given by f(x) = x2. The graph of a quadratic function will serve as the
basis for drawing the graph of any quadratic function f(x) = ax2 + bx + c. we can save some labor by the
symmetry that exists. For some example, note the following:

𝑓(−3) = 𝑓( 3 ) = 9

𝑓(−1) = 𝑓( 1 ) = 1

1 1 1
𝑓(− ) = 𝑓( ) =
2 2 4

Note: when f ( -x ) = f ( x ), the graph is said to be symmetric with respect to the x-axis

The accompanying table of values gives several ordered pair of numbers that are coordinates of points on the
graph of y = x2, when the points are located on a rectangular system and connected by a smooth curve the
graph of y = x2 is obtained.

X f ( x ) = x2
-3 9
-2 4
-1 1
0 0
1 1
2 4
3 9

The curve is called parabola and every quadratic function y = ax2 + bx + c has such a parabola as its graph. The
domain of a function is set of all real numbers.

An important feature of a parabola is that it is symmetric about a vertical line called it axis of symmetry. The
graph y = x2 is symmetric with respect to the y-axis. This symmetry is due to the fact that ( -x )2 = x2.

The parabola has a turning point, called vertex, which is located at the intersection of the parabola with its
axis of symmetry. For the preceding graph coordinate of the vertex is (0,0).
 From the graph you can see that, reading from the left to right, the
curve is “falling” down to the origin and then is “rising.” These
features are technically described as f decreasing and increasing.

F(x) = x2 increasing in [0, ∞) because for each pair x2 in this interval, if x1 < x2 then f ( x1 ) < f ( x2).
The graph of y = -x2may be obtained by multiplying each of the ordinates of y = x2 by -1. This step has the
effect of “flipping” the parabola y = x2 downward, a reflection in the axis. Since the graph of y = x2 bends
“upward”, we say that the curve is concave up, also, since y = x2 bends “downward”, we say that the curve is
concave down.

Next consider y = 2x2. It is clear from this equation that y values can be obtained by multiplying x2 by 2. So we
may take the graph of y = x2 and double or multiply its ordinate by 2 to create the points on the parabola y =
1
2x2. Similarly, to obtain the graph of 𝑦 = 2 𝑥 2 . We divide the y value by 2.
To graph the y = -2x2 you may first graph y = 2x2 as before and then draw the reflection in the x-axis, or you
may graph the y = -2x2 and then multiply by 2. Now consider quadratic function y = g(x) = ( x - 1 )2. If we write
x2 as ( x – 0 )2, then a useful comparison between these two function can be made.

Y1 = f(x) = ( x – 0 )2
Y2 = g(x) = ( x – 1 )2

Just x = 0 is the axis of symmetry for the graph of f, so x = -1 is the axis of symmetry for g. similarly, the
parabola y = ( x + 1 )2 = [ x – ( -1)]2 has x = -1 as an axis of symmetry.

More on Parabola
Let us now put several ideas together and draw the graph of this function.

Y = f(x) = ( x + 2 )2 – 2

An effective way to do this is to begin with the graph of y = x2, shift the graph 2 units to the left for y = ( x + 2
)2 – 2.

Note that the graph y = ( x + 2 )2 – 2 is congruent to the graph of y = x2. The vertex of the curve is at ( -2, -2 ),
and the axis of segments is the line x = -1. The minimum value of the function, -2, occurs at the vertex.
In general:

1. The graph of y = a( x – h )2 + k is congruent to the graph y = ax2 but is shifted h units horizontally and k
units vertically.
2. The horizontal shift is to the right if h > 0 and to the left if h < 0.
3. The vertical shift is upward if k > 0 and downward if k < 0.
4. The vertex is at ( h, k ) and the axis of symmetry is the line x = h.
5. If a < 0, the parabola opens downward and k is the maximum value.
6. If a > 0, the parabola opens upward and k is minimum value.

Example

Graph the parabola y = -2( x – 3 )2 + 4

Solution:

The graph will be a parabola congruent to y = -2x2 with vertex at ( 3, 4 ) and with x = 3 as axis of symmetry. A
brief table of values together with the graph is shown.

X 1 2 3 4 5
y -4 2 4 2 -4

REFERENCES:
Villa, N. et. al. (2010). College Algebra: Revised Edition. Pasay City: Unlad Publishing
House.

Dawkins, P. (2021). Definition of a Function. Retrieved from

https://tutorial.math.lamar.edu/classes/alg/functiondefn.aspx

CK-12. (2016). Applications of Linear Functions. Retrieved from

https://www.ck12.org/book/ck-12-algebra-i-concepts-honors/section/4.7/