In this section we examine the following question:

"Is the variable s a function of the variable t?"

The goal of this section is develop a "feel" for functional relationships,

and furthermore, do it in such a way that your understanding

of a functional relationship does not depend on the particular letters

(variables) used. (This is why I have used different letters to denote

the variables - to avoid biasing your thinking towards our special

variables x and y.)

Interpreting the Terminology

In the quotation,

"Is the variable s a function of the variable t?"

the letter s is usually y and the letter t is usually x and so the above

question becomes: "Is y a function of x?". However, it may be that

letter s refers to the letter x and the letter t means letter y, the

question now becomes: "Is x a function of y?" The letters s and t

could represent any pair variables of interest.

Suppose I make the assertion: "s is a function of t." What does this

statement mean, what are its implications?

1. The variable t is to be considered the independent variable.

2. The variable s is to be considered the dependent variable.

3. The acceptable values of the variable t vary over a set of numbers

that is referred to as the **domain** of the function.

4. The values of the function are symbolically represented by the

letter s. The values of the function come from the **range** of the

function.

5. There is some rule of association - a rule that associates with

each value of t in the domain of the function, a corresponding

value of s in the range. The rule of association my be given

explicitly or implicitly.

6. Notationally, "s is a function of t" means s = f(t).

**Quiz.**

1. Consider w as a function of z. Then, corresponding to each value

of z is only one value of w.

(a) True

(b) False

2. Consider h as a function of q. Then h is the independent variable.

(a) True

(b) False

3. Consider x a function of y. Then x may be considered a member

of the domain of the function.

(a) True

(b) False

4. Consider z as a function of x. Then, corresponding to each value

of z is only one value of z.

(a) True

(b) False

5. Consider w as a function of s. Then w may be considered a member

of the range of the function.

(a) True

(b) False

6. Consider t as a function of w. Then symbolically, this
means that

t = f(w).

(a) True

(b) False

**Passing Score: **6 out of 6.

**End Quiz.**

Let's now practice recognizing functions through a series of examples

and exercises.

**
Example 2.14.** Let x and y be real-variables. Suppose it is known

that y is related to x by the equation 2x

(a) Is y a function of x?

(b) Is x a function of y?

5y

(a) Is y a function of x?

(b) Is x a function of y?

**Exercise 2.42.** Let s and t be related to each other by way
of the

equation s - 4t + t^{2} = 1.

(a) Is s a function of t?

(b) Is t a function of s?

**Exercise 2.43.** Consider the equation x^{2} + y + 2 = 1. I wouldn't

think of asking you whether y is a function of x or whether x is a

function of y --- they are not. Let m be any number. Consider the

straight line given by y = mx and visualize the intersection of the line

y = mx with the circle x^{2} + y^{2} = 1.

(a) Is x a function of m, where x is the x-coordinate of the point(s)

of intersection between y = mx and x^{2} + y^{2} = 1.

(b) Let x be the variable described in part (a). Is m a function of

x?

**
The Vertical Line Test
**Suppose you have a curve C drawn in the xy-plane. How can we tell

whether this curve C represents y as a function of x? There is a simple

graphical test.

Vertical Line Test: A curve C in the xy-plane defines y as a function of x if it is true that every vertical line intersects the curve at no more than one point. |

**Important. **The x-axis is assumed to be the horizontal
axis, and so

the meaning of vertical is perpendicular to the x-axis.

**Exercise 2.44. **Taking the definition of** function** into consideration,

the orientation of the axes (x-axis is horizontal), and the geometry of

the graph of a curve, justify in your own mind the Vertical Line Test.

**Exercise 2.45.** Assume the usual orientation of the xy-axis system

(i.e. the x-axis is horizontal). Suppose we have a curve C in the xy-axis

plane. Under what conditions, similar to the **Vertical** Line Test,

under which we can assert that the curve defines x as a function of y?

The** Vertical **and the results of **Exercise 2.45** can be
consolidated

into a single statement which is stated independently of orientation

of the axis system.

The Function Line Test: A curve C in the xy-plane defines y as a function of x if it is true that every line perpendicular to the x-axis intersects the curve at no more than one point. |

Where, in this test, we do not assume that the x-axis is
necessarily

the horizontal axis.

The above concepts are independent of the letters used to describe

them. Here are a couple of questions using other letters.

**Exercise 2.46.** Let C be a curve in the st-plane. Under what conditions,

similar to the **Function** Line Test, under which we can assert

that

(a) the curve defines s as a function of t;

(b) the curve defines t as a function of s.

**Exercise 2.47. Quiz.
3. Graphing. First Principles**

section here discussing the fundamental principles and techniques

of graphing a function.

**4. Methods of Combining Functions**

Functions can be combined in a variety of ways to create new functions.

In this section, we discuss ways in which we can use arithmetic

operations for this purpose.

**4.1. The Algebra of Functions**

**
Equality of Functions**

Let f and g be functions. We say that f = g provided.

2. f(x) = g(x), for all

More informally, two functions are the same if they have the same

domain of definition (condition 1), and pointwise they have the same

values (condition 2).

The first example illustrates the equality of two functions. It is a two-step

method: **(1) **Check whether the domains are equal; **(2)** Check

whether the functions, pointwise, have the same values.

**Example 4.1.** Consider the following two functions:

Is it true that f = g?

The next example is almost the same as the previous one, but with

two subtle changes. The signs in the numerator and denominator of

the function g have been changed to negative signs. As in the previous

example, the numerator and denominator have a common factor,

when you cancel the common factor you get g(x) = x, this is the same

definition of f. So the two functions are equal, right?

**Example 4.2.** Are the following two functions equal?

**Example 4.3. **Consider the following two functions:

Is it true that f = g?

**Exercise 4.1. **Determine whether the two functions f and g are

equal.

**Exercise 4.2.** Determine whether the two functions f and g
are

equal.

**Scalar Multiplication**

Let f be a real-valued function of a real variable, and let k ∈ R be a

constant. define a new function, denoted kf, to be a function whose

domain is

such that,

The function kf is called a scalar multiple of f (the
constant k, in

this context, is referred to as a scalar).

Below is a sequence of simple examples.

The above discussion covered the construction of a scalar
multiple

of a function. A related topic is the recognition of scalar multiples

of functions. For example, consider the function G(x) = 2sinx, you

should recognize (or realize) that his function G is, in fact, a scalar

multiple of a more elementary function, namely the sin function. Thus,

G = 2 sin. (Looks strange doesn't it.)

This ability to recognize scalar multiples is fundamentally a very important

skill.

**
The Addition/Subtraction of Functions
**Let f and g be real-valued functions of a real variable. define f + g

to be a function whose domain is

such that

The function is called the sum of f and g. Subtraction of
two functions

is defined similarly.

**Exercise 4.3.** Study the definition of the sum of two
functions. Discuss

why the domain of f + g is defined as it is.

Let's look at some illuminating examples.

**Example 4.4. **Let and
g(x) = sin x, then the sum and

difference of f and g are

What are the domains of these functions?

**Exercise 4.4.** Let and
and let F =

f + g. Is there something strange about this definition?

The sum and difference of functions is extended to sums and differences

of many functions.

**Exercise 4.5.** Let f, g, and h be functions. define F = f + g - h.

What is the domain of F, and how do calculate the values of F?

You can combine scalar multiplication and
addition/subtraction of

functions.

**Example 4.5.** Let and

define F = 2f - 3g + 4h. Discuss the domain of F and obtain a

calculation formula for F.

Recognition. Frankly, a quite a bit more important ability is that of

recognizing sum and differences of functions. For example consider

the function.

Ideally, when you look at this function, you would, perhaps, in the

natural course of things, make a number of observations: (1) the

function is the difference of two other functions y = 6x^{2} sin(x) and

; (2) these two functions are scalar multiples of the functions

y = x^{2} sin(x) and . Here, we mentally decompose

the original function into smaller functional

"pieces."

When you look at certain problems in Calculus, for
example, this

ability to visually decompose a function this way is fundamental to

correctly analyzing the problem and successfully solving the problem.

(Of course, the functions y = x^{2} sin(x) and
can, themselves,

be broken down into smaller functional "pieces;" namely, into

- as we shall see in subsequent paragraphs

below.)

**Exercise 4.6. **Break the function

down into more elementary functions.

**
The Multiplication of Functions
**Let f and g be real-valued functions of a real variable. define fg to

be a function whose domain is

such that,

The function is called the product of f and g.

Examples in this section are much the same as in the previous section

on **addition and subtraction** of functions. Here we have an abbreviated

discussion.

One of the common examples of function multiplication is power functions.

For example, Consider the functions F(x) = x^{3}. Now it may be

convenient to think of F as a "stand-alone" function; Sometimes it

is useful to realize that F is a product of functions, which functions?

Well, define a function f(x) = x, then F(x) = f(x)f(x)f(x), for all

x ∈ Dom(F) = R. This can be written as F(x) = [f(x)]^{3}. Or, if

we want to utilize the concept of** equality of functions,** we can say:

F = f^{3}. (The f-cubed function, is the function whose value at an

x ∈ Dom(f) is given as [f(x)]^{3}.

**Exercise 4.7**. In light of the previous discussion, what can be said

about the function F(x) = (2x^{3}-1)^{5} and the function G(x) = sin^{2}(x)?

Now let's get back to a more direct illustration of the product of

functions definition.

**Example 4.6.** Consider the functions.

and h(x) = sin(x^{2}). define a new function by F = fgh. Find the

domain of definition of F, and write the calculating formula for F.

Recognition: As in the previous section, it is important to recognize

products of functions.

**Example 4.7.** Consider the function F(x) = 6x^{3} sin(x) cos^{2}(x), de-

fine the most basic of functions such that F is the product of them.

**Exercise 4.8. **Consider the function If

you wanted to write F in terms of sums, differences, and products,

of "elementary functions," what is the minimum number of distinct

functions needed to do this?

**The Quotient of Functions**

Let f and g be real-valued functions of a real variable. define f/g to

be a function whose domain is

such that,

The function is called the quotient of f and g.

Notice the domain of f/g is a bit more involved than the previous

definitions. Obviously, we cannot divide by 0, so we must ensure that

the x we use to evaluate the expression f(x)/g(x), cannot yield 0 in

the denominator; i.e. we require g(x) ≠ 0.

A skill level 0 example would be the following.

**Example 4.8.** Let f(x) = x^{3}, g(x) = (x^{2} - 1), and define F = f/g.

Discuss the natural domain of definition of F, and write a calculating

formula for F.

**Example 4.9.** Decompose the function into a

products and quotients of "elementary functions." Do a domain analysis

on same.

** Comparison of Functions**

Comparing functions is quite important in mathematics. It is very

important to understand what is meant by it.

**Exercise 4.9.** Let f and g be two given functions. Now that you have

seen a large number of definitions, can you give a good definition of

the following phrase:

"f < g over the set A."