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Honors Pre-Calculus > Module 01: Functions and their Graps > Flashcards

Module 01: Functions and their Graps Flashcards

1
Q

Determine whether the graph is the graph of a function. (2 points)

Yes

No

A

Yes

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2
Q

Determine the domain of the function. (2 points)

f(x) = √(11 - x)

  1. All real numbers except 11
  2. x > 11
  3. All real numbers
  4. x ≤ 11
A

4. x ≤ 11

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3
Q

Determine the domain of the function. (2 points)

f(x) = √(x+3) / [(x+8)(x-2)]

  1. All real numbers except -8, -3, and 2
  2. x ≥ 0
  3. All real numbers
  4. x ≥ -3, x ≠ 2
A

4. x ≥ -3, x ≠ 2

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4
Q

Find the range of the function. (2 points)

f(x) = (x + 5)2 + 8

  1. y > -8
  2. y ≥ 8
  3. y > 8
  4. All real numbers
A

2. y ≥ 8

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5
Q

Determine whether the formula describes y as a function of x. Explain your reasoning. (2 points)

y = -3x2 - 7x - 6

A

The formula “y as a function of x” means that y is the dependent variable; it depends on the independent variable (x) for its value. Therefore, the value of x will directly determine the value of y. Furthermore, if y is a function of x, then it should be able to function as a function, meaning y = -x should also be able to be written as f(x) = -x. When you look at the equation in terms of function notation, it is easier to see how y is a product of x, because whatever value is substituted for x will be equal to y.

If you look at the function when it is graphed, you will see that there is only one y output for each consecutive x value, meaning that it is a function. For example, when x = 6, f(6) = -3(6)2-7(6)-6, there is only one answer, (, -156), and the same applies for any other x value there will only be one y output. Therefore, this is a function.

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6
Q

Determine the intervals on which the function is increasing, decreasing, and constant. (3 points)

  1. Increasing x < 0; Decreasing x > 0
  2. Increasing x > 0; Decreasing x < 0
  3. Increasing x < 3; Decreasing x > 3
  4. Increasing x > 3; Decreasing x < 3
A

2. Increasing x > 0; Decreasing x < 0

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7
Q

Use the graph of f to estimate the local maximum and local minimum. (2 points)

  1. No local maximum; local minimum: approx. (1,-7.67)
  2. Local maximum: (-2,8); local minima: (-3,0) and (3,3)
  3. Local maximum: approx. (1,8.08); local minima: approx. (-2,-7.67) and (3,2.75)
  4. Local maximum: ∞ local minima: (-3,0) and (3,3)
A

3. Local maximum: approx. (1,8.08); local minima: approx. (-2,-7.67) and (3,2.75)

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8
Q
A
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9
Q

Determine algebraically whether the function is even, odd, or neither even nor odd. (2 points)

f(x) = -3x4 - 2x - 5

  1. Neither
  2. Even
  3. Odd
A
  1. Neither
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10
Q

Why is this function, f(x) = x2 + 6, an even function?

A

If the function is even, changing the x variables to be negative should result in the function being the same as it started as. In comparison, if the function is odd, changing the x variables to be negative, will result in the function being opposite of how it started.

Therefore, to prove that f(x) is an even function, f(-x) should still be equal to x2 + 6.

f(-x)=(-x)2+6

f(-x)=(x)2+6

According to this, f(-x) = f(x), meaning that the function is indeed an even function; furthermore, since the constant in the function - namely 6 - is an even number, the function is definitely even.

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11
Q

f(x) = 2x + 6, g(x) = 4x2

Find (f + g)(x). (1 point)

  1. 8x3 + 24x
  2. (2x+6)/4x2
  3. 4x2 + 2x + 6
  4. -4x2 + 2x + 6
A

3. 4x2 + 2x + 6

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12
Q

f(x) = 4x + 7, g(x) = 3x2

Find (f + g)(x). (1 point)

  1. 3x2 + 4x + 7
  2. -3x2 + 4x + 7
  3. (4x+7)/3x3
  4. 12x3 + 21x
A
  1. 3x2 + 4x + 7
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13
Q

f(x) = 4x + 6, g(x) = 2x2

Find (fg)(x). (1 point)

  1. 8x3 + 12x2
  2. 2x2 + 4x + 6
  3. 8x + 12
  4. 8x2 + 12x
A
  1. 8x3 + 12x2
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14
Q

f(x) = 4x + 7, g(x) = 3x2

Find (fg)(x). (1 point)

  1. 12x2 + 21x
  2. 3x2 + 4x + 7
  3. 12x3 + 21x2
  4. 12x + 21
A

3. 12x3 + 21x2

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15
Q

f(x) = 5x + 3; g(x) = 6x - 5

Find f/g. (1 point)

A

3

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16
Q

f(x) = 3x + 2; g(x) = 3x - 5

Find f/g. (1 point)

A

2

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17
Q

f(x) = √x+9 ; g(x) = 8x - 13

Find f(g(x)). (1 point)

  1. f(g(x)) = 2 √(2x-1)
  2. f(g(x)) = 8 √(x-4)
  3. f(g(x)) = 2 √(2x+1)
  4. f(g(x)) = 8√(x+9) - 13
A
  1. f(g(x)) = 2 √(2x-1)
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18
Q

f(x) = √x+7 ; g(x) = 8x - 11

Find f(g(x)). (1 point)

  1. f(g(x)) = 2 √(2x+1)
  2. f(g(x)) = 8 √(x+7) - 11
  3. f(g(x)) = 8 √(x-4)
  4. f(g(x)) = 2 √(2x-1)
A

4. f(g(x)) = 2 √(2x-1)

19
Q

Find f(x) and g(x) so the function can be expressed as y = f(g(x)).

y = 8x2 + 4

A
  • f(x)=x+4
  • g(x)=8x2
20
Q

Find f(x) and g(x) so the function can be expressed as y = f(g(x)). (1 point)

y = 2/x2+ 3

A
  • f(x) = x + 3
  • g(x) = 2/x2
21
Q

Find the inverse of the function.

f(x) = 5x + 6 (2 points)

A

f-1 (x) = (x-6)/5

22
Q

Find the inverse of the function.

f(x) = x3 - 4

A

f-1(x) = 3√x+4

23
Q

Find the inverse of the function.

f(x) = 4x3 - 7

A

f-1(x) = 3√[(x+7)/(4)]

24
Q

Determine if the function is one-to-one. (2 points)

  • Yes
  • No
A

Yes

25
Q

onfirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x

f(x) = 8/x and g(x) = 8/x

A

Since both f(g(x)) and g(f(x)) are eqaul to the function identity (x), they are inverse:

26
Q

Describe how the graph of y= x2 can be transformed to the graph of the given equation. (2 points)

y = x2 + 8

  1. Shift the graph of y = x2up 8 units.
  2. Shift the graph of y = x2 right 8 units.
  3. Shift the graph of y = x2 left 8 units.
  4. Shift the graph of y = x2 down 8 units.
A
  1. Shift the graph of y = x2up 8 units.
27
Q

Describe how the graph of y = x2 can be transformed to the graph of the given equation. (2 points)

y = (x-16)2

  1. Shift the graph of y = x2 down 16 units.
  2. Shift the graph of y = x2 left 16 units.
  3. Shift the graph of y= x2 right 16 units.
  4. Shift the graph of y = x2 up 16 units.
A

3. Shift the graph of y= x2 right 16 units.

28
Q

Describe how the graph of y= x2 can be transformed to the graph of the given equation. (2 points)

y = (x-10)2+8

  1. Shift the graph of y = x2 left 10 units and then down 8 units.
  2. Shift the graph of y = x2 up 10 units and then right 8 units.
  3. Shift the graph of y = x2 left 10 units and then up 8 units.
  4. Shift the graph of y = x2 right 10 units and then up 8 units.
A

4. Shift the graph of y = x2 right 10 units and then up 8 units.

29
Q

Describe how to transform the graph of f into the graph of g. (2 points)

f(x) = √x and g(x) = √-x

  1. Reflect the graph of f across the y-axis.
  2. The graph shifts up one unit.
  3. Reflect the graph of f across the y-axis and then reflect across the x-axis.
  4. Reflect the graph of f across the x-axis.
A
  1. Reflect the graph of f across the y-axis.
30
Q

Determine whether the graph is the graph of a function. (5 points)

yes or no

A

yes

31
Q

Determine the domain of the function. (5 points)

f(x) = √(9-x)

  1. x ≤ 9
  2. All real numbers except 9
  3. All real numbers
  4. x > 9
A
  1. x ≤ 9
32
Q

Find the range of the function. (5 points)

Find the range of the function. (5 points)

f(x) = x2 + 3

A

y ≥ 3

33
Q

Use the graph of f to estimate the local maximum and local minimum. (5 points)

  1. No local maximum; no local minimum
  2. Local maximum: (-2.8, 1.8); local minimum: (-1.2, 1.8)
  3. Local maximum: ∞; local minimum: -∞
  4. Local maximum: approx. (1,3.66); local minimum: approx. (4,-2.55)
A

4. Local maximum: approx. (1,3.66); local minimum: approx. (4,-2.55)

34
Q

Determine algebraically whether the function is even, odd, or neither even nor odd. (5 points)

f(x) = 143√x

  1. Neither
  2. Odd
  3. Even
A

Odd

35
Q

f(x) = √(4x + 6) and g(x) = √(4x -6)

Find (f+g)(x)

A

Answer:

√(4+6+) + √(4x-6)

36
Q

f(x) = 3x + 9, g(x) = 3x2

Find (fg)(x). (5 points)

  1. 9x + 27
  2. 9x2 + 27x
  3. 9x3 + 27x2
  4. 3x2 + 3x + 9
A

3. 9x3 + 27x2

37
Q

Find f(g(x)):

f(x) = √x+2 and g(x) = 8x -6

A

f(g(x)) = 2√(2x-1)

38
Q

Find the inverse of the function.

f(x) = 7x - 1

A

f-1 (x) = (x+1)/7

39
Q

Find the inverse of the function.

f(x) = x3 - 7

A

f-1(x) = 3√x+7

40
Q

Find the inverse of the function.

f(x) = 8x3 - 5

A

f-1(x) = 3√(x-5)/8

41
Q

Determine if the function is one-to-one. (5 points)

YES or NO

A

NO

42
Q

Describe how to transform the graph of f into the graph of g.

f(x) = x6 and g(x) = - x6

  1. Reflect the graph of f across the x-axis.
  2. Reflect the graph of f across the y-axis.
  3. Shift the graph of f down 1 unit.
  4. Reflect the graph of f across the x-axis and then reflect across the y-axis.
A
  1. Reflect the graph of f across the x-axis.
43
Q
A

Honors Pre-Calculus (5 decks)

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