College Algebra & Trigonometry, 2018a

192 CHAPTER 4. FUNCTIONS
4.2 Domain and Range of a Function
The analysis of the behavior of functions addresses questions of when the function
is increasing or decreasing, whether and where it has maximum or minimum
values, where it crosses the x or y axis, and which values of x and y aretobeincluded
in the function.
The set of values available for the x, or independent variable is called the Domain
of the function. The set of corresponding y values is called the Range of the
function.
The linear function mentioned above f(x) = 6x − 1 has a domain of all real
numbers and a range of all real numbers, x ∈ R and y ∈ R . On the other hand,
the function f(x) =x 2 has a domain of all real numbers, x ∈ R, but its range is
limited to the positive real numbers, y ≥ 0.
Considerations of the domain of a function typically refer to restrictions on which
x values will generate real number values for y. The most common restrictions
occur with the use of square root functions or rational functions.
The domain of the function f(x) = √ x − 7 would be the set of x ≥ 7, so that no
negative values are permitted under the square root. This secures the necessary
real values for y. The range for this function is y ≥ 0.
x
The domain of the function f(x) = would be x ∈ R (all real numbers),
2x − 3
but x ≠ 3 , thus avoiding a zero denominator which is an undefined value. The
2
range for this function would be y ∈ R (all real numbers), but y ≠ 1 , due to the
2
horizontal asymptote at y = 1.
2
The questions of domain and range become more interesting when considered
in relation to functions defined by graphs, or in applications. In an application
involving perimeter in which the perimeter of a rectangle is given as 50 feet, we
know that
2l +2w =50
Rewriting this as a function of w, we can say that
l = f(w) =25− w