3 ## FUNCTIONSFunctional notation: 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 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 rule that relates two variables, typically When that is the case, we say that Thus a "function" must be single-valued ("one and only one"). For example,
To each value of Domain and range The values that In the function There is one case however in which the domain must be restricted: A denominator may not be 0. In this function,
Once the domain has been defined, then the values of By the value of the function we mean the value of The range is composed of the values of the function. It is customary to call Example 1. Let the domain of a function be this set of values: A = {0, 1, 2, −2} and let the variable
a) Write the set of ordered pairs (
That is, when When b) Write the set B which is the range of the function.
Notice that to each value of
Example 2. Here is a relationship in which
When
Problem 1. Let
a) Which is the independent variable and which the dependent variable? To see the answer, pass your mouse over the colored area.
b) The domain of a function are the values of the independent variable,
b) which are the values of c) What is the natural domain of that function? Since there is no natural restriction on the values of d) The range of a function are the values of the dependent variable,
e) What is the range of that function? (Consider that the values of
(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.") f) Write any three values of that function as members of an ordered pair. For example, (1, 3), (2, 12), (3, 27) Functional notation The Say that we are considering two functions -- two rules for determining
Then it will be convenient to give each of them a name. Let us call the function -- the rule -- f." And let us call y = 5x by the name "g." We will write the following:
We read this, " The parentheses in Thus, the function
This means that the function For example,
(Lesson 18 of The function An argument x^{2} + 1 We write
"
A function of a function Again, let us consider these functions:
And now consider this function,
"
Again, Now let's look at
The parentheses in
Problem 2.
Problem 3. Let
Problem 4. Let
Problem 5. Let
Problem 6. Let
Problem 7. If
The function
Problem 8. Function of a function.
Let a) b)
Problem 9. Let
Problem 10. Let
Problem 11. This expression --
-- is called the Newton quotient or the difference quotient. Calculating and simplifying it is a fundamental task in differential calculus. For each function a)
b)
In Line 1) we squared the binomial In Line 2) we subtracted the In Line 3) we divided both the numerator and denominator by
In Line 1) we added the fractions in the numerator of the complex fraction. (Lesson 23 of In Line 2) we removed the parentheses in the numerator, and multiplied by the reciprocal of the denominator. (Lesson 22 of In Line 3) we subtracted the In Line 4) we canceled the Next Topic: Introduction to graphs Please make a donation to keep TheMathPage online. Copyright © 2016 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |