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MATH107-02 - Spring 2008 - Quiz 2 1
Name: Key
Instructions:
Put answers in the spaces provided, unless otherwise stated. Show all of your work, and show it clearly (if I can’t read it, I can’t grade it). Remember to completely simplify your answers (e.g., rationalize all denominators).
- (10 pts) Given f (x) =
x^2 − 1
and g(x) =
x + 2, find (f + g)(x), (f − g)(x), (f g)(x),
f
g
(x), and
(f ◦ g)(x). Also state the domain of each (in interval notation).
Domain of f : x 2 − 1 6 = 0 ⇒ x 2 6 = 1 ⇒ x 6 = ± 1 Domain of g: x + 2 ≥ 0 ⇒ x ≥ − 2 The domain of f + g, f − g, and f g is [− 2 , −1) ∪ (− 1 , 1) ∪ (1, ∞) The domain of f /g is (− 2 , −1) ∪ (− 1 , 1) ∪ (1, ∞)
(f + g)(x) =
x^2 − 1
x + 2 (f − g)(x) =
x^2 − 1
x + 2
(f g)(x) =
x^2 − 1
x + 2 =
x + 2
x^2 − 1
f
g
(x) =
1 x^2 − 1 √ x + 2
(x^2 − 1)
x + 2 (f ◦ g)(x) = f (g(x)) = f (
x + 2) = 1 (
√ x+2)^2 − 1
1 x+2− 1 =^
1 x+
Domain of f ◦ g is x 6 = 1 and x ≥ − 2 ⇒ [− 2 , −1) ∪ (− 1 , ∞).
(f + g)(x) = 1 x^2 − 1 +^
x + 2 Domain : [− 2 , −1) ∪ (− 1 , 1) ∪ (1, ∞)
(f − g)(x) = 1 x^2 − 1 −
x + 2 Domain : [− 2 , −1) ∪ (− 1 , 1) ∪ (1, ∞)
(f g)(x) =
√ x+ x^2 − 1 Domain :^ [−^2 ,^ −1)^ ∪^ (−^1 ,^ 1)^ ∪^ (1,^ ∞) ( f
g
(x) =
(x^2 − 1)
x + 2
Domain : (− 2 , −1) ∪ (− 1 , 1) ∪ (1, ∞)
(f ◦ g)(x) = 1 x+1 Domain :^ [−^2 ,^ −1)^ ∪^ (−^1 ,^ ∞)
- (10 pts) Let f (x) = x^4 − 4 x^2. Factor f (x) and use the factored form to find the zeros. Then sketch the graph of f (x) on the grid. Label the x− and y−intercepts.
Factored form: f (x) = x^2 (x − 2)(x + 2) Zeros: x = 0, 2 , − 2
f (x) = x 4 − 4 x 2 = x 2 (x 2 − 4) = x 2 (x − 2)(x + 2)
End behavior:
an = 1 > 0 and n = 4 (even)
⇒ y → ∞ as x → −∞, y → ∞ as x → ∞
MATH107-02 - Spring 2008 - Quiz 2 2
- (10 pts) For each of the following functions, find the inverse, if it exists. If the inverse does not exist, state why. Box your answer.
(a) f (x) =
1 + 3x
5 − 2 x
(b) g(x) = x^4 + 5
y =
1 + 3x
5 − 2 x
y(5 − 2 x) = 1 + 3x
5 y − 2 xy = 1 + 3x
5 y − 1 = 3x + 2xy
5 y − 1 = x(3 + 2y)
x =
5 y − 1
3 + 2y
y =
5 x − 1
3 + 2x
= f − 1 (x)
g(x) is not one-to-one, since if x 1 = 1 and x 2 = −1, then g(x 1 ) = 1^4 + 5 = 6 = (−1)^4 + 5 = g(x 2 ), but x 1 6 = x 2. So g−^1 (x) does not exist.
- (10 pts) Find the quotient and remainder of
x^4 + 3x^3 − 16 x^2 − 27 x + 63
x + 2
using (a) long division, and (b) synthetic division.
Long division: Synthetic division:
x + 2
x 3
- x 2 − 18 x + 9 ) x 4
- 3x 3 − 16 x 2 − 27 x + 63 x 4
- 2x 3
x^3 − 16 x^2 x^3 + 2x^2
− 18 x 2 − 27 x − 18 x^2 − 36 x
9 x + 63 9 x + 18 45
Q(x) = x 3
R(x) = 45
- (10 pts) Evaluate the following and write the result in the form a + bi. Box your answers.
(a) (3 − 4 i)(5 − 12 i) = 15 − 36 i − 20 i + 48i 2 = 15 − 56 i − 48 = − 33 − 56 i
(b)
2 − 3 i
1 − 2 i
2 − 3 i
1 − 2 i
1 + 2i
1 + 2i
2 + 4i − 3 i − 6 i 2
1 − 4 i^2
2 + i + 6
1 + 4
8 + i
5
i
(c) (−4 + i) − (2 − 5 i) = −6 + 6i
Bonus: (3 pts) i 2008 = (i 4 ) 502 = 1 502 = 1
