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MATH201 Elementary Functions > Combining Functions > Flashcards

Combining Functions Flashcards

1
Q

Define as an equation the combination of both functions:
f + g

A

f(x) + g(x) = (f + g)(x)

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2
Q

Define as an equation the combination of both functions:
f - g

A

f(x) - g(x) = (f - g)(x)

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3
Q

Define as an equation the combination of both functions:
f * g

A

f(x) * g(x) = (f * g)(x)

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4
Q

Define as an equation the combination of both functions:
f/g

A

f(x)/g(x) = (f / g)(x)

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5
Q
A
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6
Q

Write mathematically: f composed with g of x.

A
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7
Q

Write mathematically: f composed with g of 2.

A
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8
Q

Write mathematically: h composed with g of 2.

A
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9
Q

Write mathematically: f composed with p of 6.

A
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10
Q

Write mathematically: f composed with g of h of 5.

A

f(g(h(5)))

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11
Q

Write mathematically: f composed with g of h of x.

A
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12
Q

Write mathematically: c composed with g of f of x.

A
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13
Q

Write mathematically: g composed with t of f of 6.

A
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14
Q
A

3

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15
Q
A

7

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16
Q

Which of the following best approximates the value of g(h(1))?
-7, -5, 0 or 2

A

-5

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17
Q

What is f of g?

A
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18
Q

What is g(f(x))?

A

g(f(x)) = 2^2+3

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19
Q
A

1.𝑓(𝑔(0)) = 26, 𝑔(𝑓(0)) = −57
2. 𝑓(𝑔(0)) = 27, 𝑔(𝑓(0))= -94
3.𝑓(𝑔(0))= 4, 𝑔(𝑓(0))= 4
4.𝑓(𝑔(0))= 1/5 ,𝑔(𝑓(0))= 5

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20
Q
  1. Find f composed in g and simplify.
  2. Find g composed in f and simplify.
A
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21
Q
  1. Find g composed with f
A
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22
Q

What is the domain of g composed with f?

A

[0,6]

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23
Q

Why should we find the domain of a function BEFORE any simplification?

A

Because simplification could change the whole function. For example:

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24
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A
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25
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26
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27
Q
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28
Q
A
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29
Q

What are the six graphs to ALWAYS remember?

A
30
Q

What would be the formulas to shift a function up or down by c units?

A
31
Q

Give an example of vertical shifting using f(x) = x²

A
32
Q

What would be the formulas to shift a function left or right by c units?

A
33
Q

Give an example of left/right shifting using f(x) = lxl.

A
34
Q

What would be the formulas for compressing or stretching vertically a function by c units assuming c >1?

A
35
Q

Give an example of compressing or stretching using f(x) = x³.

A

y = cf(x³) stretching vertically
y = (1/c)f(x³) compressing vertically

36
Q

What would be the formulas for compressing or stretching horizontally a function by c units assuming c >1?

A
37
Q

Give an example of compressing or stretching using f(x) = mx+b.

A
38
Q

What would be the manipulation on the y = f(x) to cause a reflection over the x-axis?

A
39
Q

Give an example of reflection over the x-axis using f(x) = x².

A
40
Q

What would be the manipulation on the y = f(x) to cause a reflection over the y-axis?

A
41
Q

Give an example of reflection over the y-axis using f(x) = √x.

A

f(x)= -√x.

42
Q

Use transformations to sketch the following graph and describe the steps.

A
43
Q

Use transformations to sketch the following graph and describe the steps.

A
44
Q

Use transformations to sketch the following graph and describe the steps.

A
45
Q

If f(x) is our initial function, how would we describe the new function g(x) = f(x) + k?

A

g(x) is a vertical shift of the function f(x), where all the output values have been increased by k.

46
Q

In g(x) = f(x) + k:
If k is positive, then the graph will shift________
If k is negative, then the graph will shift _______

A

If k is positive, then the graph will shift up
If k is negative, then the graph will shift down

47
Q

A function f(x) is given as a table below. Create a table for the function g(x) = f (x) - 3

A
48
Q

If f(x) is our initial function, how would we describe the new function g(x) = f(x + k)?

A

k is a constant
then g(x) is a horizontal shift of the function f(x)

49
Q

Let f(x) be our initial function and g(x) = f(x + k) be our transformation.
If k is positive, then the graph will shift ______
If k is negative, then the graph will shift _______

A

If k is positive, then the graph will shift left
If k is negative, then the graph will shift right

50
Q

A function f(x) is given as a table below. Create a table for the function g(x) = f (x - 3):

A
51
Q

Given f (x) = l x l , sketch a graph of h(x) = f (x + 1) - 3 and find a formula for the new h function.

A
52
Q

Write a formula for this function:

A

h(x) = (√x -1) +2

53
Q

Reflect the graph below vertically and write the appropriate formula for s(x).

A

s(x) = - √x

54
Q

Reflect the graph below horizontally and write the appropriate formula for s(x).

A

s(x) = √ (-x)

55
Q

Given a function f(x), if we define a new function g(x) as g(x) = _______________
then g(x) is a ___________reflection of the function f(x), sometimes called a reflection about the ________________.

A

g(x) = - f(x)
g(x) is a vertical reflection of the function f(x), sometimes called a reflection about the x-axis

56
Q

If we define a new function g(x) as g(x) = f(- x),
then g(x) is a ______________reflection of the function f(x), sometimes called a reflection about the________.

A

g(x) = f(- x),
then g(x) is a horizontal reflection of the function f(x), sometimes called a reflection about the y-axis

57
Q

A function f(x) is given as a table below. Create a table for the function g(x) = - f (x) and h(x) = f (- x).

A
58
Q

Given a function f(x), if we define a new function g(x) as g(x) = kf (x) , where k is a constant
then g(x) is a ______________or _____________ of the function f(x).

A

Given a function f(x), if we define a new function g(x) as g(x) = kf (x) , where k is a constant
then g(x) is a vertical stretch or compression of the function f(x).

59
Q

Given a function f(x), if we define a new function g(x) as g(x) = f (kx), where k is a constant
then g(x) is a ____________________or. ______________________ of function f(x).

A

Given a function f(x), if we define a new function g(x) as g(x) = f (kx), where k is a constant
then g(x) is a horizontal stretch or compression of the function f(x).

60
Q

If k > 1, then the graph will be_________________
If 0< k < 1, then the graph will be _______________
If k < 0, then there will be ______________________________________

A

If k > 1, then the graph will be compressed by 1k
If 0< k < 1, then the graph will be stretched by 1k
If k < 0, then there will be a combination of a horizontal stretch or compression with a horizontal reflection.

61
Q

A function f(x) is given in the table below. Create a table for the function g(x) = f (1/2*x)

A
62
Q

The graph of 𝑦=𝑓(𝑥)
is given below:
On a piece of paper sketch the graph of 𝑦=𝑓(𝑥+2)
and determine the new coordinates of points A, B, and C.

A

A = (-3,0)
B = (0,1)
C = (3,0)

63
Q

The graph of 𝑦=𝑓(𝑥)
is given below:
On a piece of paper sketch the graph of 𝑦=−𝑓(𝑥)+6
and determine the new coordinates of points A, B, and C.

A

A = (-1,6)
B = (2,5)
C = (5,6)

64
Q

Let 𝑓(𝑥)= 𝑥³ + 1 and let 𝑔(𝑥)= 𝑥 + 1.
Match the functions defined below with the letters labeling their equivalent expressions.

Provide the expanded forms.
1. 𝑔(𝑥²)
2. (𝑔(𝑥))²
3. 𝑔(𝑥)𝑓(𝑥)
4. (𝑓(𝑥))²

A
  1. 1 + x²
  2. x² + 2x + 1
  3. 1 + x + x³ + x⁴
  4. 1 + 2x³ + x⁶
65
Q

Find a formula for: g(f(h(x))) =

A
66
Q

Let 𝑓(𝑥) = √(6−𝑥) and 𝑔(𝑥)= 𝑥² − 𝑥.
Then the domain of (𝑓∘ 𝑔) is equal to [𝑎,𝑏]
for a = _____ and b = _______.
Draw the graph.

A

[-2,3]

67
Q

Find the inverse of the following functions:

A
68
Q
A
69
Q

Suppose f(x) = x + 4 and g(x) = 2x -5.
Then:

A
70
Q

Consider the following function:

A
71
Q

Let f(x) = √(5 - 4x)

Find 3 decompositions of 𝑓(𝑥)=𝑝(𝑞(𝑥)) into a pair of functions 𝑝(𝑥) (the outside function) and 𝑞(𝑥) (the inside function) making the composition true.
p(x) = ______ and q(x) =_________
p(x) = ______ and q(x) =_________
p(x) = ______ and q(x) =_________

A
72
Q

Decompose the function below into 𝑢(𝑣(𝑥)). In each part, based on the function 𝑣(𝑥)
given, find the corresponding 𝑢(𝑥)
needed to decompose the function.

A

MATH201 Elementary Functions (11 decks)

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