week 1 - relations, functions and graphs

RELATIONS, FUNCTIONS AND GRAPHS

(COLLEGE ALGEBRA AND TRIGONOMETRY, Aufmann, Barker and Nation 7th ed., page

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WEEK 1 - Relations, Functions and Graphs

Two-Dimensional Coordinate System

Cartesian Coordinate Systems

• Each point on a coordinate axis is associated with a number called its coordinate.

• Each point on a flat, two-dimensional surface, called a coordinate plane or xy-plane, is associated with an ordered pair of numbers called coordinates of the point.

• Ordered pairs are denoted by (𝑥, 𝑦) , where the real numbers 𝑥 is the x-coordinate or abscissa and the real number 𝑦 is the y-coordinate or ordinate


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• The coordinates of a point are determined by the point’s position relative to a horizontal coordinate axis called the x-axis and a vertical axis called the y-axis.

• The axes intersect at the point (0,0), called the origin.

• The four regions formed by the axes are called quadrants and are numbered counterclockwise.

• The two-dimensional coordinate system is referred to as a Cartesian Coordinate System.


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Graph of an Equation by Point-Plotting

Definition of the Graph of an Equation

The graph of an equation in the two variables and is the set of all points (x, y) whose coordinates satisfy the equation.

Example 1:

Consider the equation 𝑦 = 2𝑥 + 1, graph by point-plotting.


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𝒙 𝒚 = 𝟐𝒙 + 𝟏 𝒚 (𝒙, 𝒚)

−2 2 −2 + 1 −3 (−2, −3)

−1 2 −1 + 1 −1 (−1, −1)

0 2 0 + 1 1 (0, 1)

1 2 1 + 1 3 (1, 3)

2 2 2 + 1 5 (2, 5)




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y

x

5 -4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2 -5

-3

Example 2. Graph: 𝑦 = 𝑥 + 3

Example 3. Graph: 𝑦 = 2 − 𝑥

Example 4. Graph: 𝑥2 + 𝑦 = 5

Example 5. Graph: 𝑦2 = 4𝑥


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Relations and Functions

Relation

• is referred to as any set of ordered pair.

• conventionally, It is represented by the ordered pair ( x , y ).

• x is called the first element or x-coordinate while y is the second element or y-coordinate of the ordered pair.


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Function

• a set of ordered pairs in which no two ordered pairs have the same first coordinate and different second coordinates

• every function is a relation, not every relation is a function

• it may have ordered pairs with the same second coordinate



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xy sin1

3 xy

One-to-one and many-to-one functions

Each value of x maps to only one

value of y . . .

Consider the following graphs

Each value of x maps to only one

value of y . . .

BUT many other x values map to

that y.

and each y is mapped from only

one x.

and

is an example of a one-to-one function

13 xy is an example of a many-to-one function

xy sin

xy sin1

3 xy

Consider the following graphs

and

One-to-many is NOT a function. It is just a relation. Thus a function is a relation but a relation could never be a function.

State whether the relation defines y as a function of x.

1. 𝐴 = −1, 0 , 0, 1 , 0,−1 , 3, 2 , 3, −2

2. 𝐵 = −2, 4 , −1, 1 , 0, 0 , 1, 1 , 2, 4

3. 2𝑥 − 𝑦 = 1

4. 𝑥2 + 𝑦2 = 4

5. 𝑦 = 𝑥

6. 𝑥 = 𝑦

7. 𝑦 = 9 − 𝑥

8. 𝑦 = 16 − 𝑥2

9. 4𝑦2 − 9𝑥2 =36

10. 𝑦 =𝑥+1

𝑥



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Domain and Range

Domain

• is the set of all the first coordinates of the ordered pair

Independent Variable

• is the variable that represents elements of the domain



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Range

• Is the set of all the second coordinates

Dependent Variable

• Is the variable that represents elements of the range



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Determine the domain and range of the following functions.

1. A= −2, 4 , −1, 1 , 0, 0 , 1, 1 , 2, 4

2. 2𝑥 − 𝑦 = 1

3. 𝑦 = 𝑥

4. 𝑦 = 9 − 𝑥

5. 𝑦 = 16 − 𝑥2

6. 𝑦 =𝑥+1

𝑥

7. 𝑦 =3−𝑥

𝑥−3

8. 𝑦 = 𝑥2 − 2



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Function Notation

• Functions can be named by using a letter or combination of letters, such as, 𝑓, 𝑔, 𝑕, 𝐹, 𝐺, 𝐻.

• If 𝑥 is an element of the domain of 𝑓, then 𝑓(𝑥), which is read "𝑓 of 𝑥“ or “the value of 𝑓 at 𝑥”, is the element in the range of 𝑓 that corresponds to the domain of 𝑥.

• The notation “𝑓” means the name of the function while the notation “𝑓(𝑥)“ means the value of the function at 𝑥.



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Evaluation of Functions

To evaluate a function, replace the independent variable with a number in the domain of the function and then simplify the resulting numerical expression.

Example 1:

Let 𝑓 𝑥 = 𝑥2 − 3 and evaluate:

a. 𝑓(−3)

b. 𝑓(4𝑎)

c. −2𝑓(𝑏)

d. 𝑓(𝑎 − 3)

e. 𝑓 𝑎 + 𝑓(2)



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Example 2:

Let 𝐴 𝑥 = 𝑥2 − 4, find

a. 𝐴(−2)

b. 𝐴(4)

c. 𝐴(𝑤 + 1)

Example 3:

Let 𝐺 𝑥 =𝑥

𝑥, find

a. 𝐺(−1)

b. 𝐺(2)

c. 𝐺 𝑎 , 𝑎 > 0

d. 𝐺 𝑎 , 𝑎 < 0



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Piecewise – defined Functions

• These are functions represented by more than one expression.

Example:

𝑓 𝑥 = 𝑥 − 1, 𝑥 < −3

𝑥2 − 1,−3 ≤ 𝑥 ≤ 45 − 𝑥, 𝑥 > 4



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Evaluate the following piecewise-defined functions.

1. Given: 𝑔 𝑥 = 𝑥 + 2, 𝑥 ≤ −2

4 − 𝑥2, −2 < 𝑥 ≤ 22 − 𝑥, 𝑥 > 2

,

Find each of the following

a. 𝑔(−3) b. 𝑔(0) c. 𝑔(1) d. 𝑔(3)

2. Given: 𝑕 𝑥 = 2𝑥 − 1, 𝑥 < −11 − 2𝑥, 𝑥 ≥ −1

,

Find each of the following

a. 𝑕(−2) b. 𝑕(0) c. 𝑕(3)



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Algebra of Functions

Definitions of Operations on Functions

If 𝑓 and 𝑔 are functions with domains 𝐷𝑓 and 𝐷𝑔, then

we define

a. The Sum 𝑓 + 𝑔 𝑥 = 𝑓 𝑥 + 𝑔(𝑥)

Domain: 𝐷𝑓 ∩ 𝐷𝑔

b. The Difference 𝑓 − 𝑔 𝑥 = 𝑓 𝑥 − 𝑔(𝑥)


c. The Product 𝑓 ∙ 𝑔 𝑥 = 𝑓 𝑥 ∙ 𝑔(𝑥)


d. The Quotient 𝑓/𝑔 𝑥 = 𝑓 𝑥 /𝑔(𝑥)

Domain: 𝐷𝑓 ∩ 𝐷𝑔, 𝑔(𝑥) ≠ 0



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Example 1:

Let 𝑓 𝑥 = 5𝑥 + 1 and 𝑔 𝑥 = 2 − 𝑥2. Find the following and the domain of the resulting functions:

a. 𝑓 + 𝑔 b. 𝑓 − 𝑔

c. 𝑓 ∙ 𝑔 d. 𝑓/𝑔

e. 𝑔/𝑓

Example 2:

Let 𝑕 𝑥 = 𝑥 − 1 and 𝑗 𝑥 =1

𝑥. Evaluate the following:

a. 𝑕 + 𝑗 (2) b. 𝑕

𝑗(5) c. 𝑕 ∙ 𝑗 (3)



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Difference Quotient

The expression

𝑓 𝑥+𝑕 −𝑓(𝑥)

𝑕, 𝑕 ≠ 0

is called the difference quotient of f. The expression shows the manner in which a function changes in value as the independent variable changes.



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Example 1:

Determine the difference quotient of 𝑓 𝑥 = 𝑥2 + 3.

Example 2:

The distance traveled by a ball rolling down a ramp is given by 𝑠 𝑡 = 9𝑡2, where 𝑡 is the time in seconds after the ball is released and 𝑠(𝑡) is measured in feet. Evaluate the average velocity of the ball for each time interval.

a. 2, 4 b. 2, 6 c. 2, 8



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Composition of Functions

• Composition of functions is another way in which functions can be combined.

• This method of combining functions uses the output of one function as the input for a second function.

Definition of the Composition of Two Functions

Let 𝑓 and 𝑔 be two functions such that 𝑔(𝑥) is in the domain of 𝑓 for all 𝑥 in the domain of 𝑔. Then the composition of the two functions, denoted by 𝑓 ∘ 𝑔, is the function whose

value at 𝑥 is given by 𝑓 ∘ 𝑔 𝑥 = 𝑓 𝑔 𝑥 .



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Example 1:

Consider the functions 𝑓 𝑥 = 3𝑥 + 2 and 𝑔 𝑥 = 2𝑥2 − 1. Find a. 𝑓 ∘ 𝑔 (−1) b. 𝑔 ∘ 𝑓 (−1)

Example 2:

Evaluate each composite function where 𝑓 𝑥 = 𝑥 + 2, 𝑔 𝑥 = 𝑥2 − 3𝑥, and 𝑕 𝑥 = 5 − 𝑥2.

a. 𝑔 ∘ 𝑓 4 b. 𝑓 ∘ 𝑔 −3

c. 𝑔 ∘ 𝑕 0 d. 𝑓 ∘ 𝑓 8

e. 𝑕 ∘ 𝑔 𝑘 − 1

f. Show that 𝑓 ∘ 𝑔 𝑥 = (𝑔 ∘ 𝑓)(𝑥)

g. Show that 𝑔 ∘ 𝑕 𝑥 = (𝑕 ∘ 𝑔)(𝑥)



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Example 3:

A water tank has the shape of a right circular cone with height 10 feet and radius 5 feet. Water is running into the tank so that the radius 𝑟 (in feet) of the surface of the water is given by 𝑟 = 1.5𝑡, where 𝑡 is the time (in minutes) that the water has been running.

a. The area 𝐴 of the surface of the water is 𝐴 = 𝜋𝑟2. Find 𝐴(𝑡) and use it to determine the area of the surface of the water when 𝑡 = 2 minutes.

b. The volume of the water is given by 𝑉 =1

3𝜋𝑟2𝑕. Find 𝑉(𝑡)

and use it to determine the volume of water when 𝑡 = 3 minutes.



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