Free Printable Precalculus Semester 1 Final Study

1.

Which expression represents g(f(x)) if
f(x) = 2x - 8 and
g(x) = 4x

a)

8x - 32

b)

8x² - 32x

c)

8x - 8

d)

6x - 8

2.

Given g(x)= -3x and f(x)=x²+2, find g(f(x)).

a)

g(f(x))= -3x²-6

b)

g(f(x))= -3x²+2

c)

g(f(x))=9x²+2

d)

g(f(x))= -9x²+2

3.

Given f(x) = 3x + 10
and g(x) = x - 2
Find f(g(5))

a)

19

b)

23

c)

-10

d)

None of these.

4.

a)

9 - √17

b)

4

c)

2

d)

√8

5.

Given f(x)= -3x + 7 and g(x)=2x² - 8, find g(f(x)).

a)

g(f(x))= -6x²+ 31

b)

g(f(x))= -6x²+ 24

c)

g(f(x))=18x²- 84x + 90

d)

g(f(x))=9x²- 42x - 41

6.

Find f(h(5)) based on the graphs of f(x) and g(x)

a)

-6

b)

-3

c)

6

d)

2

7.

Given j(x) = ∛(x-1)
and h(x) = x³+1,
find j(h(x)).

a)

x³+3x²-9x-3

b)

x

c)

x³-3x²+3x-1

d)

j(x) * h(x)

8.

The composition of any function, f(x), and
its inverse, f^-1(x) =

a)

x

b)

f(x)

c)

1

d)

0

9.

You can find a function's inverse algebraically by

a)

Replacing each x with y, and each y with x

b)

Evaluating f(x) for x = 0

c)

Setting f(x) = 0 and
then solving for x

d)

Replacing each x with its best friend

10.

Find $g\left(f\left(2\right)\right)$

a)

-4

b)

-2

c)

0

d)

4

11.

Find $f\left(g\left(4\right)\right)$

a)

1

b)

-2

c)

-3

d)

-4

12.

Find $f\left(g\left(-1\right)\right)$

(a)

13.

Find $g\left(f\left(6\right)\right)$

(a)

14.

Find $f\left(g\left(-1\right)\right)$

a)

-7

b)

-4

c)

0

d)

2

15.

Find $g\left(f\left(1\right)\right)$

a)

-7

b)

-1

c)

0

d)

2

16.

Find $g\left(f\left(-1\right)\right)$

(a)

17.

Find $f\left(g\left(-2\right)\right)$

(a)

18.

Find $g\left(f\left(0\right)\right)$

(a)

19.

Find $f\left(g\left(-2\right)\right)$

(a)

20.

Find the inverse of $f(x)=3x+2$

a)

$f^{-1}(x)=3x-2$

b)

$f^{-1}(x)=2+3x$

c)

$f^{-1}(x)=\frac{x-2}{3}$

21.

Find the inverse of $f(x)=2x+1$

a)

$f^{-1}(x)=2x$

b)

$f^{-1}(x)=\frac{x-1}{2}$

c)

$f^{-1}(x)=2$

22.

Find the inverse of $f(x)=4x$

a)

$f^{-1}(x)=\frac{x}{4}$

b)

$f^{-1}(x)=4$

c)

$f^{-1}\left(x\right)=\frac{1}{4}$

23.

Find the inverse of $f(x)=\frac{x-5}{3}$

a)

$f^{-1}(x)=\frac{3}{5}+x$

b)

$f^{-1}(x)=3x+5$

c)

$f^{-1}(x)=5x+3$

24.

Find the inverse of $f(x)=10+7x$

a)

$f^{-1}(x)=\frac{x-7}{10}$

b)

$f^{-1}(x)=\frac{x-10}{7}$

c)

$f^{-1}(x)=\frac{1}{10+7x}$

25.

The inverse of $f(x)=8x+30$ is $f^{-1}(x)=\frac{8}{30}$ .

a)

True

b)

False

26.

The inverse of $f(x)=x^2-1$ is $f^{-1}(x)=(x+1)^{\frac{1}{2}}$ .

a)

True

b)

False

27.

The notation for an inverse is

a)

$f^{-1}\left(x\right)$

b)

$g^{-1}\left(x\right)$

c)

All of the above

28.

are these inverse functions?

a)

no

b)

yes

c)

no way to tell

29.

What is the inverse of this function?

a)

A

b)

B

c)

C

d)

D

30.

When finding the inverse of a relation,
1st you replace f(x) with y
2nd switch the x's and y's
What do you do next?

a)

solve for y

b)

replace y with f^-1(x)

c)

solve for x

d)

find the greatest common factor

31.

Find the exact value of $\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$

a)

$-\frac{\pi}{3}$

b)

$\frac{\pi}{3}$

c)

$-\frac{\pi}{6}$

d)

$\frac{\pi}{6}$

32.

Find the exact value of $\sec^{-1}\left(-2\right)$

a)

$\frac{2\pi}{3}$

b)

$-\frac{\pi}{3}$

c)

$\frac{4\pi}{3}$

d)

$-\frac{2\pi}{3}$

33.

Q1. The principal value of $\ Sin^{-1}\left(\frac{1}{\sqrt{2}}\right)\ is\$

a)

$\frac{\pi}{4}$

b)

$\frac{\pi}{6}$

c)

$\frac{\pi}{2}$

34.

Q2. The principal value of $\ Sin^{-1}\left(\frac{1}{2}\right)\ is$

a)

$\frac{\pi}{6}$

b)

$-\frac{\pi}{4}$

c)

$-\frac{\pi}{3}$

35.

Q3. The principal value of $\ Tan^{-1}\left(-1\right)\ is$

a)

$-\frac{\pi}{4}$

b)

$-\frac{\pi}{3}$

c)

$-\frac{\pi}{6}$

36.

Which equation is true?

a)

sin (x) = 6/8

b)

cos (x) = 8/6

c)

cos (x) = 6/8

d)

tan (x) = 6/8

37.

Which equation would you use to solve for the unknown angle?

a)

x = sin^–1(4/5)

b)

x = cos^–1(4/5)

c)

x = cos^–1(5/4)

d)

x = tan^–1(4/5)

38.

Solve for x.

a)

x = 90º

b)

x = 60º

c)

x = 45º

d)

x = 30º

39.

tan(sin^-1(-1/2))

a)

√3

b)

√3/3

c)

-√3

d)

-√3/3

40.

Evaluate: cos^-1(√3/2)

a)

π/3

b)

2π/3

c)

π/6

d)

5π/6

41.

Solve
cosθ = - √(3)/2
on θ∈[0, 2π)

a)

θ = π /3, 2π /3

b)

θ = 2π /3, 4π /3

c)

θ = π /6, 5π /6

d)

θ = 5π /6, 7π /6

42.

Solve sinθ = ½ on θ∈[0, 2π)

a)

θ = π / 6, 7π / 6

b)

θ = π / 6, 5π / 6

c)

θ = 5π / 6, 7π / 6

d)

θ = 7π / 6, 11π / 6

43.

Solve for 0≤x≤2π

4cos²x- 2 = 0

a)

π/3, 5π/3

b)

π/4, 7π/4

c)

3π/4, 5π/4

d)

π/6, 11π/6

44.

Solve for 0≤x≤2π

tanx = 1

a)

π/2, π

b)

π/4, 3π/4

c)

π/4, 5π/4

d)

3π/4, 7π/4

45.

Which of the following is NOT

a solution to

tan θ = 0 ?

a)

θ = 0

b)

θ = π /2

c)

θ = π

d)

θ = 2π

46.

Find all solutions to the equation $\sin\theta\ =\ .6$ . You answer should be in degrees. Consider n to be any integer

a)

$36.9\degree\ and\ 143.1\degree$

b)

$36.9\degree\ +\ 360\degree n$

c)

$36.9\degree\ +\ 2\pi n\ \ and\ 143.1\degree\ +2\pi n$

d)

$36.9\degree\ +\ 360\degree n\ \ and\ \ 143.1\degree\ +\ 360n$

47.

Find all solutions to $\cos\theta\ =\ -.3\ \ \ for\ 0<\theta<2\pi$ . Your answer should be in radians. Assume n is any integer. Check all answers that apply.

a)

$4.408\ +\ 2\pi n$

b)

$1.266$

c)

$1.875$

d)

$1.875\ +2\pi n$

e)

$4.408$

48.

Find all solutions for 2sin(6x) + √3 = 0.

Your answer should be in radians.

a)

x = .175 + πn/3 and x = .873 + πn/3

b)

x = 1.920 + 2πn and x = 1.222 + 2πn

c)

x = .698 + πn/3 and x = .873 + πn/3

d)

x = 4.189 + 2πn and x = 5.236 + 2πn

49.

Find all solutions for 2cos(3πx) + 5 = 4 .

Your answer should be in radians.

a)

x = .278 + 2n/3 and x = .611 + 2n/3

b)

x = .111 + 2n/3 and x = .556 + 2n/3

c)

x = .222 + 2πn and x = .444 + 2πn

d)

x = .222 + 2n/3 and x = .444 + 2n/3

50.

You are standing 10 feet away from a tree. There is a cat up in the tree. Your eyes make a 48 degree angle to look up at the cat. How high is the cat?

a)

11.1 ft.

b)

9 ft.

c)

7.4 ft.

d)

13.8 ft.

51.

A 4 ft tall child creates a shadow that is 7.5 ft long. What is the angle of elevation of the sun?

a)

62 degrees

b)

49 degrees

c)

28 degrees

d)

15 degrees

52.

A ladder is leaning against a house at an angle of 39 degrees. If the ladder is 12 ft. long, how far is the bottom of the ladder from the house?

a)

15.4 ft.

b)

9.3 ft.

c)

20.2 ft.

d)

5.3 ft.

53.

Find the exact value of sin 2x if sin x = 12/13 and x is in the first quadrant.

a)

120/169

b)

25/169

c)

60/169

d)

5/13

54.

Use Sum or Difference Identities to find the exact value of each expression.

cos(75°)

a)

1/4

b)

c)

d)

55.

$\cos70^o\cos40^o-\sin70^o\sin40^o\$ is equivalent to

a)

$\cos30^o$

b)

$\cos70^{o\ }$

c)

$\cos110^o$

d)

$\sin70^{o\ }$

56.

If $\sin A=\frac{4}{5},\tan B=\frac{5}{12},$ and A and B are first quadrant angles, what is the value of $\sin\left(A+B\right)$ ?

a)

$\frac{63}{65}$

b)

$-\frac{33}{65}$

c)

$\frac{33}{65}$

d)

$-\frac{63}{65}$

57.

If $\sin\theta=\frac{\sqrt{5}}{3}$ , then $\cos2\theta$ equals

a)

$\frac{1}{3}$

b)

$-\frac{1}{3}$

c)

$\frac{1}{9}$

d)

$-\frac{1}{9}$

58.

$Find\ the\ exact\ value\ of\ \tan\ \frac{\pi}{12}$

a)

$2-\sqrt{3}$

b)

$-2+\sqrt{3}$

c)

$\sqrt{3}$

d)

$\frac{\sqrt{3}}{3}$

59.

Simplify √-4

a)

2i

b)

-2i

c)

2

d)

4i

60.

Simplify √-23

a)

-√23

b)

23i

c)

23

d)

i√23

61.

Simplify √-25

a)

i√25

b)

5i

c)

-5i

d)

25i

62.

Simplify 2√-32

a)

2i√32

b)

4i√2

c)

8i√2

d)

-16i√2

63.

$i^2=?$

a)

i

b)

-1

c)

-i

d)

1

64.

$i^{63}\ =?$

a)

i

b)

-1

c)

-i

d)

1

65.

Simplify:
-3i(4 + 2i)

a)

6 - 12i

b)

6 + 12i

c)

-12i - 6i²

d)

-12i + 6i²

66.

Simplify:
(2 - 3i)(5 + 4i)

a)

22 - 7i

b)

22 + 7i

c)

22 - 7i²

d)

22 + 7i²

67.

Simplify:
i⁷

a)

i

b)

-i

c)

1

d)

-1

68.

Simplify.

(a)

69.

What is the real portion of the number

3+5i

a)

3

b)

5

c)

-3

d)

-5

70.

What is the imaginary portion of the number $-7-12i$

a)

7

b)

12

c)

$-7$

d)

$-12$

71.

Two complex numbers are added. How do we do this?

a)

We add up all 4 numbers you see.

b)

We add up all the big numbers.

c)

We add only the real parts. Because we cannot see the imaginary parts.

d)

We add the real parts together, and then separately add the imaginary parts together. Your final answer is in the form. $a+bi$

72.

What complex number is represented by the graph?

a)

-3-i

b)

-3+i

c)

1+3i

d)

1-3i

73.

What is the radius of a unit circle?

a)

1

b)

2

c)

1/2

d)

0

74.

The hypotenuse of any right triangle from the center of the unit circle to an edge is always

a)

1

b)

sqt2/2

c)

sqt3/2

d)

1/2

75.

Based on your unit circle cos(0^o)=

a)

1

b)

0

c)

-1

d)

1/2

76.

Based on your unit circle sin(90^o)=

a)

1/2

b)

0

c)

-1

d)

1

77.

sin(7π/4)

a)

1/2

b)

-√2/2

c)

√2/2

d)

-1/2

78.

sin(3π/2)

a)

0

b)

1

c)

-1

d)

1/2

79.

Simplify the expression:
$\left(1\ +\ 5i\right)\ +\ \left(1\ -\ 5i\right)$

a)

$2\ +\ 10i$

b)

$2\ +\ 10i^2$

c)

$2$

d)

$-8$

80.

Simplify the expression:
$\left(8\ +\ 9i\right)\ +\ \left(4\ -\ 6i\right)$

a)

$17\ +\ 3i$

b)

$12\ +\ 3i^2$

c)

$17\ -2i$

d)

$12\ +\ 3i$

81.

Simplify the expression:
$\left(5\ +\ 14i\right)\ -\ \left(10\ +\ 2i\right)$

a)

$5\ -16i$

b)

$-5\ +\ 16i$

c)

$5\ -\ 12i$

d)

$-5\ +\ 12i$

82.

$2\left(3\ +\ 4i\right)-\left(4\ -\ 6i\right)$

a)

$2\ +\ 14i$

b)

$10\ +\ 2i$

c)

$-1\ -2i$

d)

$-2\ -2i$

83.

(-4i)(3i)(4i)

a)

24

b)

-48i

c)

48i

d)

60

84.

(i)(3i)

a)

-3i

b)

-3

c)

3

d)

2i

85.

(-3-i)(6-i)

a)

-21-7i

b)

-19-3i

c)

-15-5i

d)

-17-9i

86.

Determine the modulus of : $\left|3-2i\right|$

a)

$\sqrt{-13}$

b)

$\sqrt{-1}$

c)

$\sqrt{13}$

d)

$5$

87.

What is the correct set-up to find the absolute value of -9+5i?

a)

$-9+5$

b)

$\sqrt{\left(-9+5i\right)^2}$

c)

$\sqrt{\left(-9\right)^2+\left(5\right)^2}$

d)

$\sqrt{-9+5}$

88.

Convert the polar coordinates to rectangular form

a)

(1, 1)

b)

(1, -1)

c)

(-1, 0)

d)

(-1, -1)

89.

Express the complex number in polar form.
-2 + 5i

a)

5.39 ( cos 1.95 + i sin 1.95)

b)

5.39 ( cos 65.06 + i sin 65.06)

c)

2.65 ( cos 1.95 - i sin 1.95 )

d)

2.65 ( cos 65. 06 - i sin 65.06 )

90.

Express the complex number in polar form.
6 + 2i

a)

6.32 ( cos 0.32 + i sin 0.32)

b)

6.32 ( cos 0.45 + i sin 0.45)

c)

1.73 ( cos 0.32 + i sin 0.32)

d)

1.73 ( cos 0.45 - i sin 0.45)

91.

What quadrant would the point in polar form land in?
4 ( cos π /3 + i sin π /3)

a)

Quadrant 4

b)

Quadrant 1

c)

Quadrant 2

d)

Quadrant 3

92.

Which of the following is the complex conjugate of 3 + 4i?

a)

-3 + 4i

b)

-3 - 4i

c)

3 - 4i

d)

7i

93.

Express the point in polar form in rectangular form.
4 ( cos π /3 + i sin π /3)

a)

2 + 2√3 i

b)

2 + √3 i

c)

2 - 2√3 i

d)

-2 + 2√3 i

94.

How many zeros does the following function have?
f(x)= x⁵- 3x³ + x

a)

8

b)

9

c)

5

d)

3

95.

Which of these is a possible rational root of the polynomials?

x³ - 6x² - x + 30

a)

0

b)

4

c)

-9

d)

-15

96.

Which of the following roots match the above polynomial:

a)

-3,-1,4

b)

-3,2i,-2i

97.

What is the total number of roots for the following equation?
y = 4x⁶ - 12x⁵ - x⁴ + 2x³ - 6x² - 5x + 10

a)

4

b)

5

c)

6

d)

7

98.

How many solutions does the polynomial f(x)=3x⁴ + 5x² - 6x have?

a)

4

b)

2

c)

3

d)

5

99.

All imaginary roots comes in pairs.

a)

True

b)

False

100.

Factor f(x)= x³ - 2x²-13x - 10 given that
(x - 5) is one factor.

a)

(x- 5)(x - 1)(x - 2)

b)

(x - 5)(x + 2)(x - 2)

c)

(x - 5)(x + 2) (x + 1)

d)

(x - 5)(2x - 13)

Precalculus Semester 1 Final Study

Total questions: 100

Worksheet time: 3hrs 24mins

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