Math 135
Functions: Domain and Range
Solutions
1. Compute:
(a) f (x) = 2x + 1, f (2x) =
Answer 1.
f (2x) = 2(2x) + 1 = 4x + 1
(b) f (x) = x2 − 1, f (x + 1) =
Answer 2.
f (x + 1) = (x + 1)2 − 1 = x2 + 2x + 1 − 1 = x2 + 2x = x(x + 2)
(c) f (x) = 7x + 11, f (f (x) − 9) =
Answer 3.
f (f (x) − 9) =
=
=
=
=
(d) f (x) =
1
, f ( x1 )
x−1
Answer 4.
7(f (x) − 9) + 11
7(7x + 11 − 9) + 11
7(7x + 2) + 11
49x + 14 + 11
49x + 25
=
1
=
f
x
1
x
1
=
−1
1
x
1
−
x
x
=
1
1−x
x
=
x
1−x
(x)
(e) f (x) = 2x + 4, f (x+h)−f
=
h
Answer 5.
2(x + h) + 4 − (2x + 4)
2x + 2h + 4 − 2x − 4
2h
f (x + h) − f (x)
=
=
=
=2
h
h
h
h
2. Determine if the following are functions.
(a) y = 4
Answer 6. Yes, this is a horizontal line and passes the vertical line test.
(b) {(−2, 0), (3, 5), (−2, 4), (1, 5)}
Answer 7. No, because for x = −2 we have two different values.
(c) The relation which assigns to each person the month and day of their birthday.
Answer 8. Yes, each person has one birthday.
(d)
 2
 x , −3 ≤ x ≤ 0;
x,
0 < x ≤ 2;
y=
 3
x,
2 ≤ x < 5.
Answer 9. No, at x = 2 there are two different possible values: 2 and 23 = 8.
University of Hawai‘i at Mānoa
60
R Spring - 2014
Math 135
Functions: Domain and Range
Solutions
3. Find the domain and range of the following functions.
(a) y = 2
Answer 10.
Domain: (−∞, ∞)
Range: {2}
√
(b) {(−1, 0), (3, 1), (0, −1), (1, −1), (2, 2), (4, 1)}
Answer 11.
Domain: {−1, 0, 1,
√2, 3, 4}
Range: {−1, 0, 1, 2}
(c) The relation which assign to each person the first letter of their last name.
Answer 12.
Domain: The set of all people.
Range: The alphabet.
(d)

 0, −3 ≤ x ≤ 0;
x,
0 < x < 2;
g(x) =
 2
x,
2 ≤ x < 5.
Answer 13.
Domain: [−3, 0] ∪ (0, 2) ∪ [2, 5) = [−3, 5]
Range: {0} ∪ (0, 2) ∪ [4, 25) = [0, 2) ∪ [4, 25)
University of Hawai‘i at Mānoa
61
R Spring - 2014