FUNCTIONS AND RELATIONS

Given two sets A and B, we can describe the relationship between the elements from set A to set B. A

relation from A to B is a subset of the cross-product of A and B where the first components come

from set A and the second components come from set B.

Relation R ⊆ A x B = {(x,y) | x ∈ A and y ∈ B }

The domain of the relation R is the set consisting the first components of all the ordered pairs in the

relation while the range is the set consisting the second components.

D (R) = {x | (x,y) ∈ R}

R (R) = {y | (x,y) ∈ R}

A function from A to B, denoted by 𝑓: 𝐴 → 𝐵 is a relation from A to B such that for every 𝑥 ∈ 𝐴, there

is a unique 𝑦 ∈ 𝐵 such that (𝑥, 𝑦) ∈ 𝑓. That is, it is a rule that assigns each element of A exactly one

element of B.

Ways of Representing Relations

1. A set of ordered pairs

A relation from A to B is a function if no two ordered pairs have the same first component.

Example: Which of the following set of ordered pairs are functions? Determine the domain

and the range of each.

a. {(1, 𝑥), (2, 𝑦), (3, 𝑧)}

b. {(𝑎, 3), (𝑏, 4), (𝑐, 4)}

c. {(𝑥, 3), (𝑦, 5), (𝑦, 7)}

d. {(𝑟𝑒𝑑, 3), (𝑦𝑒𝑙𝑙𝑜𝑤, 6), (𝑏𝑙𝑢𝑒, 6)}

e. {((2,3), 𝑏), ((1,3), 𝑐), ((1,3), 𝑑)}

2. Mapping

One-to-one FUNCTION Many-to-one FUNCTION

One-to-many RELATION Many-to-many RELATION

3. Equation

Example: Which of the following equations represent a function?

a. {(𝑥, 𝑦)|𝑦 = 𝑥2 − 1)

b. {(𝑥, 𝑦)| 𝑦 = 2𝑥 + 4}

c. {(𝑥, 𝑦)| 𝑦 2 = 𝑥 − 2}

d. {(𝑥, 𝑦)| 𝑦 = 𝑥2)

e. {(𝑥, 𝑦)|𝑥 + 𝑦2 = 3}

(2,3)

4. Graph

A vertical line test is used to determine whether a relation is a function or not. We first plot the

given relation on the x-y plane. If a vertical line is drawn and it intersects the graph of the relation

in at most one point, then the relation is a function.

Example: Given the graph of the following relations, determine which of them is/are functions.

a. 𝑦 = 𝑥2 b. 𝑦 = 2𝑥 + 4 c. 𝑦 2 = 𝑥 − 2

Function Notation and Evaluation

If f is the function having as its domain variable x and as its range variable y, the symbol f(x) [“f of x” or “f

at x”] denotes the particular value of y that corresponds to the value of x.

Examples:

1. Given that f is the function defined by f(𝑥) = 𝑥2 + 3𝑥 + 4, find the following:

a. f(0)

b. f(2)

c. f(2h)

d. f(2x)

e. f(x+h)

f. f(x) + f(h)

(NOTE THE DIFFERENCE BETWEEN (e) and (f))

2. Given the function f defined as f(𝑥) = {2𝑥 − 1 𝑖𝑓 𝑥 ≥ 2

𝑥2 𝑖𝑓 𝑥 < 2, find:

a. f(-1)

b. f(2)

c. f(4)

Domain of some functions

Find the domain of the following functions:

1. 𝑓(𝑥) = √𝑥 + 5

2. 𝑓(𝑥) = √2 − 𝑥

3. 𝑓(𝑥) =2𝑥

𝑥−2

4. 𝑓(𝑥) =3

𝑥+5

5. 𝑓(𝑥) =3

√𝑥+5

6. 𝑓(𝑥) =4𝑥

√3−𝑥

Determining the range of the function based on its graph

1. 2.

0

5

10

15

20

-5 0 5

0

5

10

15

-5 0 5 -4

-2

0

2

4

0 5 10

(-2, 0)

-1

0

1

2

3

4

-1 0 1 2 3 4-4

-3

-2

-1

0

1

-4 -3 -2 -1 0 1 2 3 4

3. 4.

1. 5. 6.

Some Types of Functions

1. Constant function

The function f, defined by the equation

𝑓(𝑥) = 𝑐 , where c is any real number is called a CONSTANT FUNCTION.

Domain: {𝑥|𝑥 ∈ R} or (−∞, +∞) or R

Range: {𝑦|𝑦 = 𝑐} or {c}

Its graph is a horizontal line at a directed distance of c units from the x axis.

Examples:

a. The graph of 𝑓(𝑥) = 2 is as follows: b. The graph of 𝑓(𝑥) = −3 is given by:

Domain: {𝑥|𝑥 ∈ R} Domain: {𝑥|𝑥 ∈ R}

Range: {𝑦|𝑦 = 2} Range: {𝑦|𝑦 = −3}

(-2, 0)

(2,4)

(-2, -3) (-2, -4)

(-2, 1)

(3, 4)

(3, 1)

-4-3-2-101234

-3 -2 -1 0 1 2 3 4

-3-2-101234567

-4 -3 -2 -1 0 1 2 3 4

2. Linear Function

The function f, defined by the first degree equation

𝑓(𝑥) = 𝑚𝑥 + 𝑏,

where m and b are constants, and 𝑚 ≠ 0 is called a LINEAR FUNCTION.

Domain: {𝑥|𝑥 ∈ R} or (−∞, +∞) or R

Range: {𝑦|𝑦 ∈ R} or (−∞, +∞) or R

Graph of a linear function:

Its graph is a line having slope m and y-intercept b. To sketch the graph of a linear function, it is

enough to find two points that satisfy the function, and connect the points with a straight line.

Two suggested points for sketching the graph of a linear function are the x-intercept and the y-

intercept.

Definition: The zeros of the function f are the values of x for which f(x)=0. A linear function has only

one zero. This gives the x-intercept of the linear function.

Examples: Find the zeros of the linear function f, defined by the following expressions for f(x).

a. 𝑓(𝑥) = 2𝑥 + 4

b. 𝑓(𝑥) = −5𝑥 + 10

c. 𝑓(𝑥) = 10 − 12𝑥

Plot the following function on the coordinate plane, then find the domain and range.

a. 𝑦 = 3𝑥 + 2

Domain: {𝑥|𝑥 ∈ R}

Range: {𝑦|𝑦 ∈ R}

b. 𝑦 = −3𝑥 + 2

Domain: {𝑥|𝑥 ∈ R}

Range: {𝑦|𝑦 ∈ R}

Exercises: Sketch the graph of the following functions, then determine the domain and the range.

a. 𝑓(𝑥) = 2𝑥 − 7

b. 𝑓(𝑥) = 6 −𝑥

2

c. 𝑓(𝑥) = 3𝑥 + 6

d. 𝑓(𝑥) = −4𝑥 + 8

3. Quadratic Function

The general quadratic function is defined by

𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

Where a, b, c are constants representing real numbers and a ≠ 0.

Domain: {𝑥|𝑥 ∈ 𝑅} or (−∞, +∞) or R

Range: 𝑖𝑓 𝑎 > 0, {𝑦|𝑦 ≥4𝑎𝑐−𝑏2

4𝑎} or [

4𝑎𝑐−𝑏2

4𝑎, +∞)

𝑖𝑓 𝑎 < 0, {𝑦|𝑦 ≤4𝑎𝑐−𝑏2

4𝑎} or (−∞,

4𝑎𝑐−𝑏2

4𝑎]

Solution of the quadratic equation

An equation that can be written as

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0

Where a, b, c are real number constants and 𝑎 ≠ 0, is called a second-degree polynomial equation, or

quadratic equation, in the variable x. A quadratic equation written in this form is said to be in

standard form.

1. Solution by factoring

This can be applied to any quadratic equation in standard form for which the left side can be

factored.

i. Factor the left side of the equation.

ii. Set each factor equal to zero.

iii. The solution set of the given quadratic equation is the union of the solution sets of the two

linear equations.

Examples. Find the solution set of the following equations:

1. 𝑥2 + 3𝑥 − 10 = 0

2. 4𝑥2 − 5𝑥 = 6

3. (𝑥 − 4)(𝑥 + 1) = 0

2. Solution by quadratic formula

If 𝑎 ≠ 0, the solutions of the equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 are given by

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

Examples: Use the quadratic formula to find the solution set of the equation.

a. 𝑥2 + 6𝑥 − 1 = 0

b. 3𝑥2 + 5𝑥 + 3 = 0

c. 2𝑥2 = 5𝑥 − 1

Graph of a quadratic function:

The graph of a quadratic function is called a parabola. To sketch its graph:

i. Solve for the vertex (maximum or minimum point of the parabola) given by the point

(−𝑏

2𝑎,

4𝑎𝑐−𝑏2

4𝑎).

ii. Find one point with x-coordinate that is less than the x-coordinate of the vertex. Find

another point with x-coordinate greater than the x-coordinate of the vertex.

iii. Connect the 3 points from steps 1 and 2 to sketch the parabola.

Note that if a>0,the parabola opens upward and the vertex is a minimum value, and if

a<0, the parabola opens downward and the vertex is a maximum value.

Examples: Sketch the graph of the following quadratic functions, then find the domain and the range.

a. 𝑦 = 𝑥2 + 6𝑥 + 5

Here, 𝑎 = 1, 𝑏 = 6, 𝑎𝑛𝑑 𝑐 = 5

Using the formula, we get −𝑏

2𝑎= −3, 𝑎𝑛𝑑

4𝑎𝑐−𝑏2

4𝑎= −4. Hence, the vertex is at the point (-3,-4).

To complete the graph, solve for f(x) at any x value less than -3, and at any x value greater than -3, then

connect the points with the vertex. In this example, a>0, so the graph opens upward. It is given by:

Domain: {𝑥|𝑥 ∈ 𝑅}

Range: {𝑦|𝑦 ≥ −4}

b. 𝑦 = −2𝑥2 + 8𝑥 − 5

Here, 𝑎 = −2, 𝑏 = 8, 𝑎𝑛𝑑 𝑐 = −5

Using the formula, we get −𝑏

2𝑎= 2, 𝑎𝑛𝑑

4𝑎𝑐−𝑏2

4𝑎= 3. Hence, the vertex is at the point (2,3).

To complete the graph, solve for f(x) at any x value less than 2, and any x value greater than 2, then

connect the points. Note that since a<0, we will expect the graph to open downward. It is given by:

Domain: {𝑥|𝑥 ∈ 𝑅}

Range: {𝑦|𝑦 ≤ 3}

-5

-4

-3

-2

-1

0

1

2

-6 -5 -4 -3 -2 -1 0 1 2

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

-5 0 5 10

Exercises: For each of the following functions f, where f(x) is given by the following expressions, find

the maximum or minimum and draw the graph.

a. 2𝑥2 + 𝑥 + 6

b. 𝑥2 − 4𝑥 + 4

c. 𝑥2 + 6𝑥 + 11

d. 𝑥2 + 6𝑥 + 5

e. 𝑥2 + 𝑥 − 6

f. 2𝑥2 + 5𝑥 − 12

g. −2𝑥2 + 11𝑥 − 15

h. 6𝑥2 − 17𝑥 + 5

i. −2𝑥2 + 5𝑥 + 8

j. 𝑥2 + 6𝑥 + 11

k. −3𝑥2 + 5𝑥 − 4

Problems involving quadratic functions

1. A rectangular field is to be enclosed with 240 meters of fence.

a. Let x meters be the length of the field. Express the area of the field (in square meters) as a

function of x.

b. Find the dimensions of the largest rectangular field that can be enclosed with 240 meters of

fence.

2. A rectangular garden is to be fenced off with 100 ft. of fencing materials.

a. Let x feet be the length of the garden. Express the area of the garden (in square feet) as a

function of x.

b. Find the dimensions of the largest rectangular garden that can be fenced off with 100 ft of

fencing material.

3. A carpenter can construct bookcases at a cost of $40 each. If the carpenter sells the bookcases for x

dollars each, it is estimated that 300 – 2x bookcases will be sold per month. Determine the selling

price of each bookcase that will give the carpenter the greatest monthly profit. Note that

𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 − 𝐶𝑜𝑠𝑡.

4. A toy manufacturer can produce a particular toy at a cost of $10 per toy. It is estimated that if the

selling price of the toy is x dollars, then the number of toys that are sold each day is 45 – x.

Determine what the selling price of each toy should be in order for the manufacturer to realize the

maximum daily profit.

4. Piecewise function

- a function that is defined on a sequence of intervals.

- a function which is defined by multiple sub-functions, each sub-function applying to a certain

interval of the main function's domain (a subdomain).

Examples: For each of the following piecewise functions, sketch the corresponding graph, then find the

domain and the range.

a. 𝑦 = {−2 𝑖𝑓 𝑥 ≤ 32 𝑖𝑓 3 < 𝑥

b. 𝑦 = {𝑥 + 3 𝑖𝑓 𝑥 ≠ 32 𝑖𝑓 𝑥 = 3

c. 𝑦 = {2𝑥 − 1 𝑖𝑓 𝑥 < 2

0 𝑖𝑓 𝑥 ≥ 2

d. 𝑦 = {𝑥2 𝑖𝑓 𝑥 ≠ 27 𝑖𝑓 𝑥 = 2

e. 𝑦 = {𝑥2 − 4 𝑖𝑓 𝑥 < 32𝑥 − 1 𝑖𝑓 3 ≤ 𝑥

f. 𝑦 = {𝑥 − 2 𝑖𝑓 𝑥 ≤ 0

𝑥2 + 1 𝑖𝑓 0 < 𝑥

g. 𝑦 = {3𝑥 + 2 𝑖𝑓 𝑥 ≠ 18 𝑖𝑓 𝑥 = 1

h. 𝑦 = {𝑥2 − 4 𝑖𝑓 𝑥 ≠ 3−2 𝑖𝑓 𝑥 = 3

i. 𝑦 = {1 − 𝑥2 𝑖𝑓 𝑥 < 03𝑥 + 1 𝑖𝑓 0 ≤ 𝑥

j. 𝑦 = {6𝑥 + 7 𝑖𝑓 𝑥 ≤ −24 − 𝑥 𝑖𝑓 − 2 < 𝑥