Question

Hey, everyone, I'm here to give a small tutorial on functions in calculus. Rate and CNC, please. As always, enjoy.

-Regards, Compassionate

Answer 1

What is a function?
WHEN ONE THING DEPENDS on another, as for example the area of a circle depends on the radius -- in the sense that when the radius changes, the area also will change -- then we say that the first is a "function" of the other. The area of a circle is a function of -- it depends on -- the radius.


Mathematically:
A relationship between two variables, typically x and y,
is called a function
if there is a rule that assigns to each value of x
one and only one value of y.
When that is the case, we say that y is a function of x.
Thus a "function" must be single-valued ("one and only one").

For example,
y = 2x + 3.
To each value of x there is a unique value of y.
The values that x may assume are called the domain of the function. We say that those are the values for which the function is defined.
In the function y = 2x + 3, the domain may include all real numbers. x could be any real number. Or, as in Example 1 below, the domain may be arbitrarily restricted.
There is one case however in which the domain must be restricted: A denominator may not be 0. In this function,
y = 1
x − 2 ,

x may not take the value 2. For, division by 0 is an excluded operation. (Lesson 5 of Algebra.)
Once the domain has been defined, then the values of y that correspond to the values of x, are called the range. Thus if 5 is a value in the domain of
y = 2x + 3, then 13 (2· 5 + 3) is the corresponding value in the range.

By the value of the function we mean the value of y. And so when x = 5, we say that the value of the function y = 2x + 3, is 13.
The range is composed of the values of the function.
It is customary to call x the independent variable, because we are given, or we must choose, the value of x first. y is then called the dependent variable, because its value will depend on the value of x.


Example 1. Let the domain of a function be this set of values:
A = {0, 1, 2, −2}
and let the variable x assume each one. Let the rule that relates the value of y to the value of x be the following:
y = x² + 1.


a) Write the set of ordered pairs (x, y) which "represents" this function.
Answer. {(0, 1), (1, 2), (2, 5), (−2, 5)}
That is, when x = 0, then y = 0² + 1 = 1.
When x = 1, then y = 1² + 1 = 2. And so on.


b) Write the set B which is the range of the function.
Answer. B = {1, 2, 5, 5}. The values in the range are simply those values of y that correspond to each value of x.
Notice that to each value of x in the domain there corresponds one -- and only one -- value of the function. Even though the value 5 is repeated, it is still one and only one value.
Example 2. Here is a relationship in which y is not a function of x:
y² = x


When x = 4, for example -- y² = 4 -- then y = 2 or −2. To each value of x, there is more than one value of y.
Problem 1. Let y be a function of x as follows:
y = 3x²
a) Which is the independent variable and which the dependent variable?


x is the independent variable, y is the dependent.

b) The domain of a function are the values of the independent variable, b) which are the values of x.

c) What is the natural domain of that function?
Since there is no natural restriction on the values of x, the natural domain of that function is any real number. x could take any value on the x-axis.

d) The range of a function are the values of the dependent variable,
d) which are the values of y.

e) What is the range of that function? (Consider that the values of x² are
e) never negative.)
y ≥ 0
(If you are not viewing this page with Internet Explorer 6 or Firefox 3, then your browser may not be able to display the symbol ≥, "is greater than or equal to;" or ≤, "is less than or equal to.")

Answer 2

Are you going to do a tutorial on calculus?

Answer 3

Yes.