FUNCTIONS – Zed Vitals

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FUNCTIONS

Test your understanding of functions with this short quiz! Covering key concepts like function notation, domain and range, types of functions, and basic problem-solving, this quiz is perfect for reinforcing your knowledge and identifying areas that need a quick refresher.

1 / 92

Which of the following is a function?

{(1, 2), (2, 3), (3, 4)}

{(1, 2), (1, 3), (2, 4)}

{(1, 2), (2, 2), (3, 2)}

Both b and c

2 / 92

A relation is a function if:

Every input has at least one output

Every input has exactly one output

Every output has exactly one input

It passes the vertical line test

3 / 92

Which relation is not a function?

y = x^{2}

x = y^{2}

y = \sqrt{x}

y = ∣ x ∣

4 / 92

The domain of

$f (x) = \frac{1}{x}$

is:

All real numbers

x > 0

R ∖ {0}

x \geq 0

5 / 92

The range of

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

is:

[0, \infty)

(- \infty, \infty)

R

(0, \infty)

6 / 92

A function is injective (one-to-one) if:

Every output has at least one input

It passes the horizontal line test

Every output has exactly one input

It is symmetric about the y-axis

7 / 92

The slope of

$f (x) = - 3 x + 5$

is:

-3

-5

8 / 92

The function

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

has a vertical asymptote at:

x = 0

y = 0

x = - 2

x = 2

9 / 92

The domain of

$f (x) = \sqrt{x - 4}$

is:

x \geq 4

All real numbers

x > 4

x \leq 4

10 / 92

The graph of $f (x) = x^{3}$ is:

A parabola

Symmetric about the origin

A straight line

Symmetric about the y-axis

It is an odd function ( $f (- x) = - f (x)$ ).

11 / 92

The function

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

has a horizontal asymptote at:

y = 2

y = 3

y = 0

y = - 1

12 / 92

The y-intercept of

$f (x) = \frac{x^{2} - 9}{x + 3}$

is:

(0, 3)

(0, -3)

Does not exist

(0, 0)

13 / 92

The range of

$f (x) = \sqrt{9 - x^{2}}$

is:

[0, 3]

[- 3, 3]

(- \infty, 3]

[0, \infty)

14 / 92

The function

$f (x) = \frac{x + 2}{x^{2} - 4}$

simplifies to:

\frac{1}{x^{2} - 4}

Undefined

\frac{1}{x + 2}

\frac{1}{x - 2}

15 / 92

The graph of $f (x) = ∣ x + 2 ∣$ is shifted:

Right by 2 units

Down by 2 units

Left by 2 units

Up by 2 units

16 / 92

The domain of

$f (x) = \log (x - 3)$

is:

x > 3

All real numbers

x \geq 3

x < 3

17 / 92

The range of

$f (x) = e^{x}$

is:

[0, \infty)

(0, \infty)

R

(- \infty, \infty)

18 / 92

The graph of $f (x) = \frac{1}{x}$ has:

No asymptotes

A vertical asymptote at x=0

Both vertical and horizontal asymptotes

A horizontal asymptote at y=0

19 / 92

The value of $∣ - 5 ∣$ is:

-5

Undefined

20 / 92

The solution to

$∣ x ∣ = 3$

is:

x = 3

No solution

Both x=3 and x=-3

x = - 3

21 / 92

The domain of

$f (x) = \frac{1}{x^{2} - 4}$

is:

x > 2

R ∖ {- 2, 2}

R ∖ {0}

x \geq 2

22 / 92

The range of

$f (x) = \sin x$

is:

[0, 1]

R

[- 1, 1]

(- \infty, \infty)

23 / 92

The domain of

$f (x) = \sqrt{4 - x^{2}}$

is:

[- 2, 2]

(- 2, 2)

(- \infty, 2]

[2, \infty)

24 / 92

The range of

$f (x) = x^{2} + 2$

is:

[2, \infty)

(2, \infty)

R

(- \infty, 2]

25 / 92

The domain of

$f (x) = \tan x$

excludes:

x = \frac{π}{2} + n π

(n ∈ ℤ)

x = n π

(n ∈ ℤ)

All real numbers

x = 0

26 / 92

The range of

$f (x) = \frac{1}{x}$

is:

(0, \infty)

R ∖ {0}

[0, \infty)

R

27 / 92

The domain of

$f (x) = \ln (x + 1)$

is:

x \geq - 1

x < - 1

x > - 1

x \leq - 1

28 / 92

The graph of

$f (x) = e^{x}$

approaches:

y = \infty

x \to \infty

None of the above

y = 0

x \to - \infty

Both approaches

29 / 92

The graph of

$f (x) = \sin x$

is:

Symmetric about the origin

Both periodic and symmetric

Periodic with period

2 π

Neither periodic nor symmetric

30 / 92

The graph of $f (x) = ∣ x - 2 ∣$ has a vertex at:

(0, 0)

(-2, 0)

(2, 0)

(0, 2)

31 / 92

The graph of $f (x) = \log_{2} x$ passes through:

(2, 1)

(0, 1)

(1, 0)

Both (1,0) and (2,1)

32 / 92

The graph of $f (x) = x^{2} - 4$ is a parabola opening:

Upwards

Downwards

Left

Right

33 / 92

The graph of $f (x) = \frac{x}{x - 1}$ has a hole at:

x=0

x=1

y=1

No holes

34 / 92

The graph of $f (x) = \cos x$ has a maximum at:

x=π

x=0

x=π/2

x=-π

35 / 92

The solution to $∣ x - 2 ∣ \leq 3$ is:

-1 ≤ x ≤ 5

x ≤ -1 or x ≥ 5

x ≥ 5

x ≤ -1

36 / 92

The inequality $∣ 2 x + 1 ∣ > 5$ holds for:

x > 2

x < -3

Both x>2 and x<-3

-3 < x < 2

37 / 92

The solution to $∣ x + 4 ∣ \geq 2$ is:

x ≥ -2

x ≤ -6

-6 ≤ x ≤ -2

Both x≥-2 and x≤-6

38 / 92

The inequality $∣ 3 x - 6 ∣ < 9$ is equivalent to:

x < -1 or x > 5

-1 < x < 5

x > 5

x < -1

39 / 92

The solution to $∣ 5 - x ∣ > 1$ is:

x < 4 or x > 6

4 < x < 6

x > 6

x < 4

40 / 92

The inequality

$∣ 2 x ∣ \leq 8$

simplifies to:

x ≤ 4

x ≥ -4

-4 ≤ x ≤ 4

x ≥ 4

41 / 92

The solution to

$∣ x + 1 ∣ < 0$

is:

All real numbers

x = -1

x < -1

No solution

42 / 92

The range of

$f (x) = - ∣ x ∣$

is:

(- \infty, 0]

[0, \infty)

R

(- \infty, \infty)

43 / 92

The function $f (x) = ∣ x - 3 ∣$ is shifted:

Left by 3 units

Right by 3 units

Up by 3 units

Down by 3 units

44 / 92

The equation

$∣ 2 x - 1 ∣ = 5$

has solutions:

x = 3

x = -2

x = 2

Both x=3 and x=-2

45 / 92

The graph of

$f (x) = ∣ x + 2 ∣ - 1$

has its vertex at:

(2, -1)

(-2, 1)

(-2, -1)

(2, 1)

46 / 92

The function

$f (x) = ∣ x^{2} - 4 ∣$

has critical points at:

x = ±2

x = 0

x = 4

No critical points

47 / 92

The solution to

$∣ 3 x + 1 ∣ < 4$

is:

x < -5/3 or x > 1

-5/3 < x < 1

x > 1

x < -5/3

48 / 92

The range of

$f (x) = ∣ x ∣ + 2$

is:

(2, \infty)

R

(- \infty, 2]

[2, \infty)

49 / 92

The graph of

$f (x) = ∣ x ∣$

is:

A straight line

A parabola

A V-shape

A hyperbola

50 / 92

The solution to

$∣ x - 2 ∣ \leq 3$

is:

x ≤ -1 or x ≥ 5

-1 ≤ x ≤ 5

x ≥ 5

x ≤ -1

51 / 92

The inequality

$∣ 2 x + 1 ∣ > 5$

holds for:

x > 2

x < -3

Both x>2 and x<-3

-3 < x < 2

52 / 92

The solution to

$∣ x + 4 ∣ \geq 2$

is:

x ≥ -2

x ≤ -6

-6 ≤ x ≤ -2

Both x≥-2 and x≤-6

53 / 92

The inequality

$∣ 3 x - 6 ∣ < 9$

is equivalent to:

-1 < x < 5

x < -1 or x > 5

x > 5

x < -1

54 / 92

The solution to

$∣ 5 - x ∣ > 1$

is:

4 < x < 6

x > 6

x < 4 or x > 6

x < 4

55 / 92

The inequality

$∣ 2 x ∣ \leq 8$

simplifies to:

x ≥ -4

-4 ≤ x ≤ 4

x ≤ 4

x ≥ 4

56 / 92

The solution to

$∣ x + 1 ∣ < 0$

is:

All real numbers

x = -1

x < -1

No solution

57 / 92

The inverse of

$f(x) = 3x – 2$

is:

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

f^{-1}(x) = frac{x – 2}{3}

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

Does not exist

Swap

$x$

and

$y$

, then solve for

$y$

58 / 92

A function has an inverse if it is:

Even

Odd

Bijective

Periodic

59 / 92

The inverse of

$f(x) = x^2$

(for

$x geq 0$

) is:

f^{-1}(x) = sqrt{x}

f^{-1}(x) = -sqrt{x}

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

Does not exist

60 / 92

The inverse of

$f(x) = e^x$

is:

f^{-1}(x) = ln x

f^{-1}(x) = log x

f^{-1}(x) = e^{-x}

Does not exist

61 / 92

The function

$f(x) = sin x$

(restricted to

$[-frac{pi}{2}, frac{pi}{2}]$

) has an inverse:

f^{-1}(x) = cos^{-1} x

f^{-1}(x) = tan^{-1} x

f^{-1}(x) = sin^{-1} x

Does not exist

62 / 92

The inverse of

$f(x) = frac{1}{x}$

is:

f^{-1}(x) = x

f^{-1}(x) = frac{1}{x}

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

Does not exist

63 / 92

To find the inverse of

$f(x) = frac{2x + 1}{x – 3}$

, you would:

Swap

x

and

y

, then solve for

y

Differentiate

Integrate

Take the logarithm

64 / 92

The inverse of

$f(x) = log_2 x$

is:

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

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

f^{-1}(x) = frac{1}{log_2 x}

Does not exist

65 / 92

The function

$f(x) = x^3 + 1$

has an inverse because it is:

Strictly increasing

Even

Periodic

Not continuous

66 / 92

The inverse of

$f(x) = frac{x + 1}{x – 1}$

is:

f^{-1}(x) = frac{x + 1}{x – 1}

f^{-1}(x) = frac{x – 1}{x + 1}

f^{-1}(x) = frac{1 – x}{x – 1}

Does not exist

Solve

$y = frac{x + 1}{x – 1}$

for

$x$

67 / 92

$f(x) = 2x$

and

$g(x) = x + 3$

, then

$f circ g(x)$

is:

2x + 3

2x + 6

x + 6

2x^2 + 6x

$f(g(x)) = 2(x + 3) = 2x + 6$

68 / 92

$f(x) = frac{1}{x}$

and

$g(x) = x + 1$

, then

$g circ f(x)$

is:

frac{1}{x + 1}

frac{1}{x} + 1

x + frac{1}{x}

frac{x + 1}{x}

$frac{1}{x} + 1$

69 / 92

The composition

$f circ g$

is defined only if:

The range of

g

is a subset of the domain of

f

f

and

g

are both linear

g

is invertible

f

is differentiable

$f$

must accept outputs of

$g$

70 / 92

$f(x) = e^x$

and

$g(x) = ln x$

, then

$f circ g(x)$

is:

x

e^x

ln(e^x)

Undefined

$e^{ln x} = x$

for

$x > 0$

71 / 92

$f(x) = x^2$

and

$g(x) = sin x$

, then

$g circ f(x)$

is:

sin^2 x

sin(x^2)

x^2 sin x

(sin x)^2

$sin(x^2)$

72 / 92

$f circ g(x) = frac{1}{x + 1}$

and

$g(x) = x – 1$

, then

$f(x)$

is:

frac{1}{x}

frac{1}{x – 1}

frac{1}{x + 2}

x + 1

$f(x) = frac{1}{x + 2}$

73 / 92

$f(x) = 3x + 2$

and

$g(x) = frac{x – 2}{3}$

, then

$f circ g(x)$

is:

x

3x

x + 4

frac{x}{3}

$f(g(x)) = 3left(frac{x – 2}{3}right) + 2 = x$

74 / 92

The composition

$f circ f^{-1}(x)$

always equals:

0

1

x

Undefined

75 / 92

The vertex of

$f(x) = x^2 – 4x + 3$

is at:

(2, -1)

(-2, 1)

(1, 0)

(0, 3)

Complete the square:

$(x – 2)^2 – 1$

76 / 92

The roots of

$f(x) = 2x^2 – 8x + 6$

are:

x = 1

and

x = 3

x = -1

and

x = -3

x = 2

and

x = 4

No real roots

Solve

$2x^2 – 8x + 6 = 0$

using factoring.

77 / 92

The solution to

$x^2 – 4 > 0$

is:

x < -2

x > 2

-2 < x < 2

x > 2

x < 2

The parabola opens upwards with roots at

$x = pm 2$

78 / 92

The axis of symmetry of

$f(x) = -x^2 + 6x – 5$

is:

x = 3

x = -3

x = 1

x = 5

The axis of symmetry is

$x = -frac{b}{2a} = 3$

79 / 92

The maximum value of

$f(x) = -2x^2 + 4x + 1$

is:

3

1

-1

5

The vertex form gives the maximum at

$f(1) = 3$

80 / 92

The quadratic function with roots at

$x = 2$

and

$x = -1$

is:

f(x) = x^2 – x – 2

f(x) = x^2 + x – 2

f(x) = x^2 – 2x + 1

f(x) = x^2 + 2x + 1

Use factored form:

$(x – 2)(x + 1) = x^2 – x – 2$

81 / 92

The discriminant of

$f(x) = 3x^2 – 6x + 2$

is:

12

0

-12

6

$D = b^2 – 4ac = 12$

82 / 92

The graph of

$f(x) = x^2 + 4x + 5$

opens:

Upwards

Downwards

Horizontally

Has no direction

The coefficient of

$x^2$

is positive

83 / 92

The range of

$f(x) = x^2 – 6x + 9$

is:

[0, infty)

(-infty, 0]

[9, infty)

(-infty, 9]

The function is

$(x – 3)^2$

, always non-negative

84 / 92

The quadratic function

$f(x) = ax^2 + bx + c$

has a minimum if:

a > 0

a < 0

b = 0

c = 0

85 / 92

The solution to

$x^2 – 5x + 6 = 0$

is:

x = 2

and

x = 3

x = -2

and

x = -3

x = 1

and

x = 6

No real solutions

Factor to

$(x – 2)(x – 3) = 0$

86 / 92

The inequality

$x^2 + 2x – 3 leq 0$

holds for:

-3 leq x leq 1

x leq -3

x geq 1

x leq 1

x geq -3

87 / 92

The solution to

$2x^2 – 5x + 2 < 0$

is:

frac{1}{2} < x < 2

x < frac{1}{2}

x > 2

x < 2

x > frac{1}{2}

88 / 92

The inequality

$x^2 – 6x + 9 geq 0$

is true for:

x = 3

only

All real numbers

x geq 3

x leq 3

89 / 92

The solution to

$-x^2 + 4x – 3 > 0$

is:

1 < x < 3

x < 1

x > 3

x > 3

x < 1

90 / 92

The inequality

$3x^2 – 2x – 1 leq 0$

holds for:

-frac{1}{3} leq x leq 1

x leq -frac{1}{3}

x geq 1

x leq 1

x geq -frac{1}{3}

91 / 92

The solution to

$x^2 + x + 1 > 0$

is:

All real numbers

x > -1

x < -1

No solution

92 / 92

The inequality

$x^2 – 9 leq 0$

is true for:

-3 leq x leq 3

Your score is

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