Domain and Range: One Explicit Example (3.8.2)

Question 1

• The domain of a function is the set of real numbers that can be used for x.

Answer 1

• The domain of a function is the set of real numbers that can be used for x.

Question 2

• Values of x must be excluded from the domain if they create either a denominator equal to zero or a negative value under
a square root.

Answer 2

• Values of x must be excluded from the domain if they create either a denominator equal to zero or a negative value under
a square root.

Question 3

• The range of a function is the set of real numbers that are possible for y. The range of a function depends on the function’s
domain.

Answer 3

• The range of a function is the set of real numbers that are possible for y. The range of a function depends on the function’s
domain.

Question 4

• R is the notational symbol for the set of all real numbers.

Answer 4

• R is the notational symbol for the set of all real numbers.

Question 5

In this function, x is multiplied by 2 and then 5 is added to
that product. To determine the function’s domain, consider
what values of x, if any, must be excluded. Since any number
can be multiplied by 2 and there is no denominator or square
root to be considered, there are no real numbers that must be
excluded from the domain. Therefore, the domain of the
function f(x) = 2x + 5 is the set of all real numbers.
The set of all real numbers can be written as a compound
inequality, –∞ 0. One method for determining
the range is to examine the the values of f(x) that result from
the x-values in each of these three inequalities. For this simple
function it is apparent that any value of f(x) is possible.
Therefore, the range for this function is also the set of all real
numbers. However, the range is not always all real numbers
when the domain is all real numbers.
This function represents a line that is continuous throughout
the plane (i.e, the line extends infinitely in the positive
direction and in the negative direction).
The graph visually demonstrates the fact that every real
number will be an x-value in some point on this graph.
Likewise, every real number will be a y-value in some point
on this graph.
Thus, the conclusion that the domain and range of this
function are each the set of all real numbers is visually
confirmed through this graph.

Answer 5

In this function, x is multiplied by 2 and then 5 is added to
that product. To determine the function’s domain, consider
what values of x, if any, must be excluded. Since any number
can be multiplied by 2 and there is no denominator or square
root to be considered, there are no real numbers that must be
excluded from the domain. Therefore, the domain of the
function f(x) = 2x + 5 is the set of all real numbers.
The set of all real numbers can be written as a compound
inequality, –∞ 0. One method for determining
the range is to examine the the values of f(x) that result from
the x-values in each of these three inequalities. For this simple
function it is apparent that any value of f(x) is possible.
Therefore, the range for this function is also the set of all real
numbers. However, the range is not always all real numbers
when the domain is all real numbers.
This function represents a line that is continuous throughout
the plane (i.e, the line extends infinitely in the positive
direction and in the negative direction).
The graph visually demonstrates the fact that every real
number will be an x-value in some point on this graph.
Likewise, every real number will be a y-value in some point
on this graph.
Thus, the conclusion that the domain and range of this
function are each the set of all real numbers is visually
confirmed through this graph.