Finding the Domain and Range - Del Mar College
East Campus, CB 117
361-698-1579
Math Learning Center
West Campus, HS1 203
361-698-1860
FINDING THE DOMAIN & RANGE
Definition of Domain: the set of all possible x-values which will make the function "work", and
will give real y-values.
Example: ??(??) = {(3,5), (?2,7), (8,0)} The x values make up the domain. The domain is {-2, 3, 8}.
The values are arranged in numerical order.
HOW TO FIND THE DOMAIN:
1. Radicals of even root: the radicand must be a positive or zero, so to find what x can be, set
the radicand to ¡Ý zero
y = ¡Ìx ? 4
y = ¡Ìx 2 + 7x + 12
x?4¡Ý0
x 2 + 7x + 12 ¡Ý 0
x¡Ý4
(x + 3)(x + 4) ¡Ý 0
Domain is [4, ¡Þ)
Domain is (?¡Þ,?4] ¡È [?3,¡Þ)
2. Rational Expressions: the denominator can never equal zero, so set the denominator to
zero to find what x cannot be
3x+1
f(x) =
x?2
x?2=0
x = 2 ¡ so x cannot be 2
Domain is all #¡¯s except 2
Interval is (?¡Þ, 2) ¡È (2, ¡Þ)
3. The domain is (?¡Þ, ¡Þ) for the following:
4
g(x) = 2 ?9
x
x2 ? 9 = 0
(x ? 3)(x + 3) = 0
x = 3 or x = ?3 ¡ so x cannot be 3 or -3
Domain is all #¡¯s except ?3 and 3
The interval is (?¡Þ, ?3) ¡È (?3,3) ¡È (3, ¡Þ)
a) any linear function, such as ??(??) = 3?? + 7
b) any quadratic equation, such as ?? = ?? 2 + 2?? ? 3
c) any polynomial equation where x is within the absolute value bars, such as ?? = |?3?? + 7|
3
d) any equation where x is under a radical with an odd root, such as ?? = ¡Ì?? ? 6
f) any linear inequality, such as ?? > ?? + 8
g) any polynomial function, such as ?? = 4?? 3 + 6?? 2 ? 5
4. Looking at the graph, examine the x-values from left to right
Domain is [2, ¡Þ)
Domain is (-¡Þ,2) U (2,¡Þ)
East Campus, CB 117
361-698-1579
Math Learning Center
West Campus, HS1 203
361-698-1860
Definition of Range: the set of all possible y-values the relation can produce from the x-values.
Examples: ??(??) = {(3,5), (?2,7), (8,0)} The y values make up the range. The range is {0, 5, 7}.
The values are arranged in numerical order.
HOW TO FIND THE RANGE:
1. The easiest way is to look at the GRAPH, examine the y-values from bottom to top
Range is (-¡Þ,¡Þ)
Range is (-¡Þ,0) U (0,¡Þ)
Range is [-5,¡Þ)
Range is (-¡Þ,5]
2. Algebraically: There is no set way to find the range algebraically. However, one strategy
that works most of the time is to find the domain of the inverse function (if it exists). First,
swap the x and y variables everywhere they appear in the equation and then solve for y. Find
the domain of this new equation and it will be the range of the original.
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