This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.Objectives of this module: be able to identify the independent and dependent variables of an equation, be able to specify the domain of an equation.
Overview
- Independent and Dependent Variables
- The Domain of an Equation
Independent and dependent variables
Independent and dependent variables
In an equation, any variable whose value can be freely assigned is said to be an
independent variable. Any variable whose value is determined once the other values have been assigned is said to be a
dependent variable. Two examples will help illustrate these concepts.
- Consider the equation
. If we are free to choose values for
, then
would be considered the independent variable. Since the value of
depends on the value of
,
would be the dependent variable.
- Consider the equation
. If we are free to choose values for both
and
, then
and
would be considered independent variables. Since the value of
depends on the values chosen for
and
,
would be the dependent variable.
The domain of an equation
Domain
The process of replacing letters with numbers is called numerical evaluation. The collection of numbers that can replace the independent variable in an equation and yield a meaningful result is called the
domain of the equation. The domain of an equation may be the entire collection of real numbers or may be restricted to some subcollection of the real numbers. The restrictions may be due to particular applications of the equation or to problems of computability.
Sample set a
Find the domain of each of the following equations.
, where
is the independent variable.
Any number except 0 can be substituted for
and yield a meaningful result. Hence, the domain is the collection of all real numbers except 0.
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, where
is the independent variable and the equation relates time,
, and distance,
.
It makes little sense to replace
by a negative number, so the domain is the collection of all real numbers greater than or equal to 0.
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, where the independent variable is
.
The letter
can be replaced by any real number except 4 since that will produce a division by 0. Hence, the domain is the collection of all real numbers except 4.
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, where the independent variable is
.
We can replace
by any real number and the expression
is computable. Hence, the domain is the collection of all real numbers.
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Practice set a
Find the domain of each of the following equations. Assume that the independent variable is the variable that appears in the expression on the right side of the "
" sign.
, where this equation relates the distance an object falls,
, to the time,
, it has had to fall.
all real numbers greater than or equal to 0
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Exercises
For the following problems, find the domain of the equations. Assume that the independent variable is the variable that appears in the expression to the right of the equal sign.
Exercises for review