Taibah University, Preparatory Year
MATH 101, First Semester 1435/36 (2014/15)
Revision
Choose the correct answer (a,b,c or d)
1. For f (x) = x2 − 64 and g(x) = x − 8, determine the domain of ( fg )(x).
(a) (−∞, ∞)
(c) (−∞, 8] ∪ [8, ∞)
(b) (−∞, 8) ∪ (8, ∞)
(d) [8, ∞)
√
2. For h(x) = 3x − 2 + 1, find two functions f (x) and g(x) such that
h(x) = (f ◦ g)(x)
√
√
(c) f (x) = x + 1, g(x) = 3x − 2
(a) f (x) = x + 1, g(x) = 3x − 2
√
√
(b) f (x) = x, g(x) = 3x − 2
(d) f (x) = 3x − 2, g(x) = x + 1
3. For f (x) = x2 + 2 and g(x) =
√
(a) (g ◦ f )(x) = 3x + 1
√
(b) (g ◦ f )(x) = 3x2 + 5
3x − 1, find (g ◦ f )(x)
√
(c) (g ◦ f )(x) = 3x2 − 5
√
(d) (g ◦ f )(x) = 3x2 + 7
4. For the functions f (x) = 4x2 − 6x + 5 and g(x) =
(f − g)(x) is
(a) (−∞, ∞)
(c) (7, ∞)
(b) [7, ∞)
(d) (−∞, 7)
5. For the functions f (x) =
1
x
and g(x) =
5
x−6
√
x − 7, the domain of
find the domain of (g ◦ f )(x)
(a) (−∞, 0] ∪ [0, 6] ∪ [6, ∞]
(c) (−∞, 0] ∪ [0, 61 ] ∪ [ 16 , ∞]
(b) (−∞, 0) ∪ (0, 6) ∪ (6, ∞)
(d) (−∞, 0) ∪ (0, 16 ) ∪ ( 16 , ∞)
6. If f (x) =
1
x−1
and g(x) =
3x−10
x+2 ,
then domain of (f og)(x) is
(a) (−∞, −2) ∪ (−2, 6) ∪ (6, ∞)
(c) (−∞, −6) ∪ (−6, 2) ∪ (2, ∞)
(b) (−∞, 2) ∪ (2, 6) ∪ (6, ∞)
(d) (−∞, −6) ∪ (−6, −2) ∪ (−2, ∞)
1
7. Given f (x) = 2x − 1 and g(x) =
√
x − 1, find (f + g)(1)
(a) (f + g)(1) = −1
(c) (f + g)(1) = 1
(b) (f + g)(1) = 0
(d) (f + g)(1) = 2
√
√
8. If f (x) = x x, g(x) = x2 , then the value of (gof )( 2) is
√
√
(c) 2
(a) ( 2)3
√
(b) 2
(d) ( 2)2
9. Given f (x) = x2 + x and g(x) = 4 − x, find (g ◦ f )(3)
(a) (g ◦ f )(3) = 8
(c) (g ◦ f )(3) = 2
(b) (g ◦ f )(3) = −8
(d) (g ◦ f )(3) = −2
10. Let f (x) = x3 − 3 and g(x) =
x2 −1
x−2 .
The value of (f g)(2) is
(a) 2
(c) undefined
(b) 25
(d) 17
11. Given f (x) =
(f + g)(x)
√
7 − x and g(x) =
√
x + 3, determine the domain of
(a) (−3, 7)
(c) (−∞, 7]
(b) [−3, ∞)
(d) [−3, 7]
12. Determine the domain of ( fg )(x), where f (x) =
√
x + 2 and g(x) = x2 − 9
(a) (−3, 2) ∪ (2, ∞)
(c) (−3, 3) ∪ (3, ∞)
(b) [−2, 2) ∪ (2, ∞)
(d) [−2, 3) ∪ (3, ∞)
√
13. For the functions f (x) = x5 + 3 and g(x) = 5 x − 3 find h(x) = (f ◦ g)(x)
√
(c) h(x) = x
(a) h(x) = 5 x − 3 + 3
√
√
(b) h(x) = 5 x − 6
(d) h(x) = 5 x + 6
2
√
4 − x and g(x) =
√
(a) (f g)(x) = 16 − x2
√
(b) (f g)(x) = x2 + 16
14. For f (x) =
√
x + 4, find (f g)(x)
√
(c) (f g)(x) = 16 + 8x − x2
√
(d) (f g)(x) = 16 − 8x − x2
15. Find the slope of the secant line through the points (x, f (x)) and
(x + h, f (x + h)), h > 0 on the graph of the function f (x) = 1 − x2
(a) m = 2x + h
(c) m = −2x + h
(b) m = −(2x + h)
(d) m = 2x − h
16. Let f (x) = x2 +x+1. Find and simplify the expression for the difference
(x)
quotient f (x+h)−f
, h 6= 0
h
(a) −(h + 2x + 1)
(c) h − 2x + 1
(b) h − 2x − 1
(d) h + 2x + 1
17. The vertex of the graph of the quadratic function f (x) = 2x2 + 16x + 25
is:
(a) (4, −7)
(c) (4, 7)
(b) (−4, 7)
(d) (−4, −7)
18. The axis of symmetry of the graph of the quadratic function
g(x) = 4x2 − 12x + 3 is:
(a) x =
(b) x =
3
2
−3
2
(c) x =
(d) x =
2
3
−2
3
19. If f (x) = −x2 + 4x − 5, then
(a) f (x) = (x − 2)2 − 1
(c) f (x) = −(x − 2)2 + 1
(b) f (x) = −(x − 2)2 − 1
(d) f (x) = −(x + 2)2 − 1
20. State the equation of the quadratic function whose graph is shown
3
(a) f (x) = 4(x + 1)2 + 4
(c) f (x) = −4(x − 1)2 + 4
(b) f (x) = −4(x + 1)2 + 4
(d) f (x) = 4(x + 1)2 − 4
21. State the equation of the quadratic function whose graph is shown
(a) f (x) = x2 − 6x − 5
(c) f (x) = x2 − 6x + 5
(b) f (x) = x2 + 6x − 5
(d) f (x) = x2 + 6x + 5
22. Select the graph of the quadratic function f (x) = −2x2 − 4x + 6
(a)
(b)
(c)
(d)
23. The graph of the quadratic function f (x) = ax2 + bx + c, a 6= 0 has two
x − intercepts if
(a) b2 + 4ac = 0
(b) b2 − 4ac > 0
(c) b2 − 4ac < 0
(d) b2 + 4ac > 0
4
24. Find the interval on which the quadratic function f (x) = x2 + 2x + 7 is
increasing.
(a) (−∞, 2]
(c) [−1, ∞)
(b) (−∞, −4]
(d) [6, ∞)
25. Find the interval on which the quadratic function f (x) = −2x2 + 4x + 5
is decreasing.
(a) [1, ∞)
(c) (−∞, 3]
(b) (−∞, 2]
(d) [7, ∞)
26. Use synthetic division and remainder theorem to evaluate f ( 31 ), where
f (x) = 3x3 + 11x2 + 2x − 16
(a) f ( 31 ) = 18
(c) f ( 31 ) = −18
(b) f ( 31 ) = −14
(d) f ( 31 ) = 14
27. The quotient of
x4 +3x2 +29x−21
x+2
is
(a) x3 − 2x2 + 7x − 15
(c) x3 − 2x2 + 7x + 15
(b) x3 + 2x2 + 7x + 15
(d) x3 − 2x2 − 7x − 15
28. The remainder of
2x3 +7x2 −x+26
x−2
is
(a) 68
(c) 86
(b) 0
(d) 19
29. Divide
12+2x3 −9x−4x2
x−3
3
x−3
(c) 2x2 + 2x − 3 −
(b) 2x2 + 2x − 3 + (3)
(d) 2x2 − 2x − 3 −
(a) 2x2 + 2x − 3 +
5
3
x−3
3
x−3
30. Use the factor theorem to determine which one of the following is a
factor of
f (x) = x3 − 6x2 + 3x + 10
(a) x + 5
(c) x + 2
(b) x − 5
(d) x − 1
31. Use the factor theorem to determine which one of the following is a
zero of
f (x) = x2 − 4x + 5
(a) −2
(c) 2
(b) 2i
(d) 2 − i
32. Determine the one-to-one function
(a) {(5, −1), (3, 2), (10, −2), (1, −1), (4, 0)}
(b) {(0, −5), (−2, −2), (5, −6), (−4, −3), (3, −10)}
(c) {(−6, 2), (4, −9), (0, 11), (−3, 7), (4, 2)}
(d) {(−5, 5), (2, 3), (0, 5), (−1, −7), (3, 0)}
33. Which of the following graph is the graph of one-to-one function
(a)
(b)
(c)
(d)
6
34. Find the inverse function of f (x) =
function
(a) f −1 (x) =
(b) f −1 (x) =
(b) f −1 (x) =
x3
2
x3
2
x 6= −2 which is a one-to-one
(c) f −1 (x) =
1−x
2−x
1+2x
1−x
(d) f −1 (x) =
35. The inverse function of f (x) =
(a) f −1 (x) =
x−1
x+2 ,
√
3
1−x
2−x
x+2
x−1
2x − 7 is
−7
(c) f −1 (x) =
+7
(d) f −1 (x) =
x3 +7
2
x3 −7
2
36. The inverse function of f (x) = x3 + 1 is
√
√
(a) f −1 (x) = 3 x − 1
(c) f −1 (x) = 3 x + 1
√
√
(b) f −1 (x) = x − 1
(d) f −1 (x) = x + 1
37. Find the inverse function of the one-to-one function:
f (x) = {(0, 1), (−2, 3), (−5, −2), (7, 9), (1, 4)}
(a) f −1 (x) = {(1, 0), (3, −2), (−5, −2), (9, 7), (1, 4)}
(b) f −1 (x) = {(0, 1), (−2, 3), (−2, −5), (9, 7), (4, 1)}
(c) f −1 (x) = {(1, 0), (3, −2), (−2, −5), (7, 9), (4, 1)}
(d) f −1 (x) = {(1, 0), (3, −2), (−2, −5), (9, 7), (4, 1)}
38. Given f (x) =
√
3
x + 6 is a one-to-one function find f −1 (2) and (f (2))−1
(a) f −1 (2) = (f (2))−1 = 2
(b) f −1 (2) = 21 , (f (2))−1 = 2
(c) f −1 (2) = 2, (f (2))−1 =
(d) f −1 (2) = (f (2))−1 =
1
2
1
2
39. Which of the following functions is one-to-one function:
√
(c) h(x) = x2 + 6x + 8
(a) f (x) = 9 − x2
√
(b) g(x) = 4
(d) r(x) = 3 x + 1 − 2
Dr.Gehan Ahmed
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