Differential Calculus 201-NYA-05
Vincent Carrier
Domain and Range
Consider the function f whose graph is given below.
y 6
d
c
-
0
a
b
x
The domain D of a function f is defined as
D = {x ∈ R : f (x) is defined}
and its range R is defined as
R = {f (x) : x ∈ D}.
For the function f above, it is clear that D = [a, b] and R = [c, d].
A function f is defined at x if f (x) can be evaluated. There are mainly two circumstances
that can prevent the evaluation of f (x): a division by zero and the eventh root of a
negative number. Thus,
(
D = all x ∈ R except where
division by 0
p
n
negative number
)
occurs.
n even
Examples: Find the domain D of f .
√
3
x5 − 1
a) f (x) = 2
x +1
Since none of the two above-mentioned problems occurs, D = R.
b) f (x) =
x+3
x−2
c) f (x) =
x2 + 1
x2 − 5x − 14
D = R \ {2}
x2 + 1
(x + 2)(x − 7)
f (x) =
d) f (x) =
√
D = R \ {−2, 7}
3 − 2x
3 − 2x ≥ 0
3 ≥ 2x
D = (−∞, 3/2]
3/2 ≥ x
x−1
e) f (x) = √
4
4x − 3
4x − 3 > 0
D = (3/4, ∞)
4x > 3
x > 3/4
f) f (x) =
√
6
x2 − x − 2
x2 − x − 2 ≥ 0
(x + 1)(x − 2) ≥ 0
D = (−∞, −1] ∪ [2, ∞)
y
6
−1
0
2
x
1
g) f (x) = √
1 − x2
1 − x2 > 0
(1 − x)(1 + x) > 0
y
D = (−1, 1)
6
−1
0
1
x