Math 1314 – College Algebra
Section 3.2 Domain and Range
Recall: Think of the domain as the set of input values (the set of all possible x-values) and the range as the set
of output values (the set of all possible y-values). The function f tells us what to do to the input value. Think
of f as a process or operation.
Summary for Domain of Polynomial, Radical, and Rational Functions:
Look for fractions with variables in the denominator. Solve denominator 6= 0.
Look for even roots with variables in the radicand. Solve radicand ≥ 0.
Form:
anything
f (x) =
x stuff
√
even
x stuff
f (x) =
.
4x + 3
x2 − x − 6
(b) f (x) = 6x + 1
(c) g(x) =
Solve x stuff 6= 0
Solve x stuff ≥ 0
Recall: The radicand is the part of the function that is under the root.
Ex: Find the domain of the function
(a) y =
To find domain:
√
2 − 3x
.
Math 1314 OS
Section 3.2 Continued
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(d) y = − √
.
x+3
(e) h(x) =
√
3
x+2
√
5−x
(f) j(x) = 2
x − 16
(g) m(x) =
√
√
x−3+ 2+x
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Math 1314 OS
Section 3.2 Continued
We can find the domain and range of functions using graphs.
To find domain, project curve onto the x-axis.
To find range, project curve onto the y-axis.
Domain:
Range:
Constant Function
Domain:
Range:
Identity Function
Domain:
Range:
Squaring Function
Domain:
Range:
Cubing Function
Domain:
Range:
Absolute Value Function
Domain:
Range:
Square Root Function
Domain:
Range:
Cube Root Function
Domain:
Range:
Reciprocal Function
Domain:
Range:
Reciprocal Squared Function
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Math 1314 OS
Section 3.2 Continued
Ex: Find the domain and range of the function from the graph.
(a)
(b)
(c)
Ex: Given the graph of y = f (x) below, answer the following questions.
(a) Find the domain of f . Find the range of f .
(b) Find the x-intercepts and y-intercept. Write as points.
(c) Find the zeroes of f .
(d) Find f (x) < 0.
(e) Find f (3).
(f) Find the number of solutions to f (x) = −2.
(g) Find the intervals where f is increasing, constant, and decreasing.
(h) Find the local minimums and local minimums, if any exist.
(i) Find the maximum and minimum, if it exists.
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Math 1314 OS
Section 3.2 Continued
Ex: Find the domain and graph:
2
x + 2x if x ≤ −1
(a) f (x) =
3 − x if x > −1
 2
 x + 2x if x < −1
5
if x = −1
(b) g(x) =

3 − x if x > −1
(c) Find f (−2), f (−1), f (7), g(−5), g(−1) and g(11)
Practice lots of graphing BY HAND!!
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