# Cartesian Coordinate System and Functions

### 8.2. Function

In the world of Mathematics one of the most common creatures encountered

is the function. It is important to understand the idea of a

function if you want to gain a thorough understanding of *Algebra and
Calculus.*

Science concerns itself with the discovery of physical or
scientific truth.

In a portion of these investigations, researchers (or engineers) attempt

to discern relationships between physical quantities of interest. There

are many ways of interpreting the meaning of the word “relationships,”

but in these lessons we are most often concerned with functional

relationships. Roughly speaking, a functional relationship between

two variables is a relationship such that one of the two variables

has the property that knowledge of it (or knowledge of its value) implies

a knowledge of the value of the other variable.

For example, the physical quantity of area, A, of a circle
is related

to the radius of that circle, r. Indeed, it is internationally known

that A = πr^{2}—an equation, I’m sure, you have had more than one

occasion to examine in the past. The simple equation A = πr^{2} sets

forth the principle of a functional relationship: Given knowledge of the

value of one variable (the independent variable), r, then we have total

knowledge of the value of the other variable (the dependent variable),

A. This causal (or deterministic) relationship one variable has with

another variable is the essence of a functional relationship.

This only difference between the example of the previous
paragraph

and any other example of a function, either one taken from the applied

fields or one that is of a more “purelyabstract” nature, is the way in

which the functional relationship is defined, and the complexity of that

definition. There are many, many ways of defining (or describing) a

functional relationship between one variable (or a set of variables) and

another variable (or another set of variables). Some of these methods

are rather “natural,” which you will encounter as you continue with

these lessons; others are “unnatural,” but we will not encounter them

at this level of play.

Before we continue with this discussion, perhaps it is
best to have a

formalized definition of a function—in the next section.

**• The Definition
**Definition. Let A be a set and B be a set. A function, f, from A into

B is a rule that associates with each element in the set A a unique

corresponding element in the set B. In this case, we write symbolically,

Definition Notes: The set A is called the domain of the
function f.

Typically in Algebra and Calculus, the set A will be an** interval** of the

real number line R. As a notation, we shall refer to the domain of the

function f by Dom( f).

■ The set B is called the codomain of f. The set B may not
be

the **range **of the function.

■ Let the elements of A are referred to by the letter x, and
those of

the set B by y. The symbol x is called the independent variable of

f, and y is called the dependent variable. The independent variable

can take on any value in the domain, Dom(f), of f.

■ A function f is a rule that associates with each element,
x, in

the set A, a unique corresponding element, y, in the set B. The usual

way we define a function is byan equation that states the relationship

between the variable x and the variable y.

■ For example, let the function f associate the number x
with the

number y, where y = x^{2}. We write,

f: x → y where y = x^{2}

or, more simply

f: x → x^{2}

thus,

f: 2 → 4

f:−3 → 9

f: 0 → 0

Where, f: 2 → 4 states that f associates with x = 2 the
unique

corresponding number y = 4.

■ The above notation is used frequently at higher levels of
mathematics,

at our level, we use the standard functional notation. Rather

than writing f: x → x^{2}, we simply write f(x) = x^{2}.

f(x) = x^{2} means f: x → x2.

Particular evaluations are carried out as follows:

f(2) = 2^{2} = 4

f(−3) = (−3)^{2} = 9

f(0) = 0^{2} = 0

More examples are given below.

■ Given a particular x in Dom(f), y = f(x) is called a value
of

the function f. For the function f(x) = x^{2}, since f(2) = 4, we can

saythat 4 (y = 4) is a value of f. It should be clear to you that −4

in not a value of the function f(x) = x^{2}.

■ For any given function, f, some numbers are values of f
while

others are not. The set of all values of a given function is called its

range, denoted by Rng (f); thus,

Rng(f) = { y | y is a value of f }

To be a value, a ‘y’ must be of the form f(x) for some x.
Thus,

Rng(f) = { y | y = f(x) for some x in Dom(f) }

■
For the function f(x) = x^{2}, the range is

Rng(f) = [0,+∞)

Do you understand why?

All the functions that we encounter in **Algebra** (and most of those

encountered in **Calculus**) defined using algebraic expression in one

unknown. For example, the expression x^{2} − 4x + 1 is an algebraic

expression in x. We can use this expression to define a function by

f(x) = x^{2} − 4x + 1

Here are a few examples of functions defined this way. This method is

*by no means the only way of defining functions.* Read these examples

completely and carefully.

**Illustration 1.** Examples of functions and numerical evaluations.

(a) Define f by f(x) = x^{2} + x. Then,

f(3) = 3^{2} +3 = 12

f(−3) = (−3)^{2} + (−3) = 6.

(b) Define g by Then,

The names of functions are determined by the user, that’s you

and me. I choose a name of g this time.

(c) Define h by Then,

Now I have changed the letter used to denote the independent

variable. I have used t instead of the traditional x—this causes

no problems I hope? Anyletter (or symbol) can be used for the

independent variable, and anyletter (or symbol) can be used

for the dependent variable.

(d) Any symbol you say? How about defining a function W by

W(ø) = ø^{3}. Then,

W(−5) = −125

W(3) = 27

Here, I have used the Greek letter ø (“phi”) for the name of the

independent variable.

**Evaluation Tip.** To evaluate a function, such as f(x) = x^{2} − 2x, at

a particular value of x = −1, first replace the independent variable

x with the number −1, then evaluate the expression. Thus, f(−1) =

(−1)^{2} − 2(−1) (replace x by −1), then evaluate f(−1) = 1+2 = 3.

Note the use of the parentheses: this is necessary because we are

replacing a single letter x by a compound symbol −1. Not to include

these parentheses (1) is mathematically and notationally *wrong, and*

(2) *invites evaluation errors.*

**Exercise 8.10.** Evaluate each of the functions defined below at the

indicated values. Passing is 100%.

(a) f(x) = 2x^{2} − 3x; f(2), f(−2), f(−1/2)

(b) g(s) = s(s + 1)(s + 2); g(0), g(1), g(−1), g(−3)

(c) h(1), h(−2), h(1/2), h(−1/2)

(d) D(w) = D(9); D(1/9)

**• For those Who want Greater Insight. Models for
Functions.**

Listed behind this link is a description of several ways in which we

can view a function. These points of view mayhelp you to understand

this important mathematical object.

**• The Domain of a Function
**In the examples in the previous paragraphs, nothing was mentioned

concerning the domains of the functions considered. In this section

we briefly discuss methods of computing the domain of a function.

The domain of the function defined is either (1)

*explicitly specified or*

*is*(2)

*not explicitly specified.*(That seems reasonable.)

**Illustration 2**. Examples of functions with explicitly specified domains.

(a) Define a function f by

f(x) = x^{2}, x≥ 1

Here, we are explicitlydefining the domain of f to be

Dom(f) = [1,+∞) = { x | x ≥ 1 }

For this function, f(2) = 4 is defined, but f(0) is not because

x = 0 does not fall into the specified domain.

(b) Define a function g by

0 ≤ x < 1

Here, we have specified the domain of g to be

Dom(g) = [0, 1 ) = { x | 0 ≤ x < 1 }

**Illustration Notes: **Such (artificial) restriction of the domains may

arise from physical considerations. Perhaps these functions, f and g

above, are modeling some physical system; within the context of this

physical system, it only make sense to consider x ≥ 1, in the case of

the function f, and 0 ≤ x < 1 in the case of g.

When the domain of a function at definition time is left unspecified,

that usually means we are to take as the domain the so-called natural

domain of the function.

*The Natural Domain of a function.
*Given a function y = f(x). The

natural domain of f is the set of all *real numbers*,* x*, for which the

value f(x) can be calculated as a *real number.*

The next example illustrates the reasoning and methods used to calculate

the natural domain of a function. **Read carefully!**

**Example 8.2. **Compute the natural domain of each of the following.

(a) f(x) = x^{2} + 3x + 1

**Strategy. **Let y = f(x) be a function, where f(x) is some algebraic

expression. The natural domain consists of all x for which . . .

• the denominator (if any) is not equal to zero; and

• anyradicands of even roots (if any) are nonnegative.

Here’s another example that incorporates all components of the above

strategy.

**Example 8.3. **Find the natural domain of

Quite typically, the **strategy** involves setting up some constraints or

conditions on the values of the independent variable in the form of

inequalities. Once you identify these inequalities, you solve them (possibly using

the Sign Chart Method). The natural domain is then the

set of all values of the independent variable that satisfy all the constraints

or conditions.

**Exercise 8.11. **Compute the natural domain of each of the following.

**• Points of Intersection of Curves
**We have seen in the previous section that determining the natural

domain of a functions oftimes require the setting up and solving of

inequalities. To determine where two curves intersect, if at all, we must

be able to set up and solve equations. This is why the basic mechanics

of solving inequalites and equations are so important—equations

and inequalities are the natural way n which we ask questions and

the techniques of solutions are the way we are able to answer these

questions.

*Determining the x-intercept.* Let y = f(x) be a function. The *xintercept,*

if there is one, is that value of x such that f(x) = 0. As

you know, every function has a graph—graphing will be taken up in

**Lesson 9**—and in terms of the graph the x-intercept is the location

on the x-axis where the graph crosses the x-axis.

**Procedure. **To find the x-intercept(s) of the function y = f(x), set up

the equation

f(x) = 0 (5)

and solve for x.

**Exercise 8.12.** Find the x-intercept(s), if any, of each of the following

functions.

(a) f(x) = 4x − 1

(b) f(x) = x^{2} − 3x + 2

(c) f(x) = x^{3} + 6x^{2} ++8x

(d) f(x) = x^{2} + x + 1

(e) f(x) = 2x^{2} − x −1

f)f(x) = x^{2} + x − 3

**Exercise 8.13. **What does the problem of finding the x-intercept of

a function have to do with the title, “Points of Intersection of Curves,”

of this section?

*Determining the intersection of two Curves. *Consider the two functions

y = f(x) and y = g(x). We wish to find all points, if any, on the

intersection of the graphs of f and g.

Let (x_{0}, y_{0}) be a point that is on both graphs of f and g. This means

f(x_{0}) = y_{0} and g(x_{0}) = y_{0}.

At this point, we have f(x_{0}) = y_{0} = g(x_{0}). This represents a criterion

for finding the points of interection of two graphs.

**Procedure. **Let y = f(x) and y = g(x) be two function. Set up the

equation

f(x) = g(x) (6)

and solve for x.

Finding the points of intersection is essentially a problem in solving

equations.

**Example 8.4**. Find the points of interection beween

(a) f(x) = 3x + 2 and g(x) = 5x − 4

(b) f(x) = x^{2} − 3x + 1 and g(x) = 2x − 5

Notice that the points of intersection can be calculated without reference

to the graph of the functions. At our level of play, finding the

points of intersection is purely an exercise in algebra.

**Exercise 8.14.** Find the cartesian coordinates of the points of intersections

of each of the following pairs of functions.

(a) f(x) = 6x + 3 and g(x) = 2x − 7

(b) f(x) = x + 3 and g(x) = 2 − 8x

(c) f(x) = x^{2} + 7x − 1 and g(x) = 4x − 3

Now for some exercises that require the use of the** Quadratic Formula.**

**Exercise 8.15.** Find the abscissas of intersection of each of the following

pairs of functions.

(a) f(x) = 2x^{2} − 5x + 2 and g(x) = x + 3

(b) f(x) = x^{2} + 4x − 1 and g(x) = 1 − 4x − x^{2}

(c) f(x) = 3x^{2} + 1 and g(x) = x^{2} − 5x

Some curves do not intersect. Investigate these kind.

**Exercise 8.16.** Verify algebraically that each pair of functions do

not intersect.

(a) f(x) = 4x − 2 and g(x) = 4x + 12

(b) f(x) = 2x^{2} + 3 and g(x) = x^{2} − 1

(c) f(x) = x^{2} + 2x + 2 and g(x) = x + 1

(d) f(x) = x^{2} − 2x − 4 and g(x) = 4x^{2} − 3

We have come to the end of Lesson 8. Congratulations of reaching

this far. In **Lesson 9, **we take up the topics of linear and quadratic

functions as well as some graphing topics.