Week 8 Flashcards

1
Q

Graph the following showing all steps in the transformation from the starting graph.

a) 𝑦=|π‘₯+4|βˆ’1

start with 𝑦=π‘₯+4

A
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2
Q
  1. Graph the following showing all steps in the transformation from the starting graph.
    b) 𝑦=3βˆ’|π‘₯βˆ’2|

start with 𝑦=π‘₯βˆ’2

A

(b) We have been asked to draw a graph of y= 3βˆ’|xβˆ’2|.

  • Start with a graph of y=|x|(blue)
  • Reflect across the x-axis: y=βˆ’|x|(green)
  • Move the graph 2 to the right: y=βˆ’|xβˆ’2|(red)
  • Move the graph up 3: y=βˆ’|xβˆ’2|+ 3 (black)
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3
Q
  1. Graph the following showing all steps in the transformation from the starting graph.
    c) 𝑦+2+|π‘₯+1|=0

start with 𝑦=π‘₯+1

A

(c) We have been asked to draw a graph ofy+ 2 +|x+ 1|= 0.

  • Start with a graph of y=|x|(blue)
  • Reflect across the x-axis: y=βˆ’|x|(green)
  • Move the graph 1 to the left: y=βˆ’|x+ 1|(red)
  • Move the graph down 2: y=βˆ’|x+ 1|βˆ’2 (black)
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4
Q
  1. Sketch the following piecewise functions, stating the domain and range.
    a) 𝑓(π‘₯)= {π‘₯βˆ’2 π‘₯<βˆ’1

{π‘₯+2 π‘₯β‰₯βˆ’1

A

a) 𝑓(π‘₯)= {π‘₯βˆ’2 π‘₯<βˆ’1

{π‘₯+2 π‘₯β‰₯βˆ’1

Domain: All real numbers OR βˆ’βˆž< x OR (βˆ’βˆž,∞)

Range:f(x)OR (βˆ’βˆž,βˆ’3) βˆͺ [1,∞)

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5
Q

2. Sketch the following piecewise function

a) 𝑓(π‘₯)= {π‘₯βˆ’2 x <βˆ’1

{π‘₯+2 π‘₯β‰₯βˆ’1

i) 𝑓(βˆ’3)

A

𝑓(βˆ’3) =βˆ’3βˆ’2 =βˆ’5

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6
Q

2. Sketch the following piecewise function

a) 𝑓(π‘₯)= {π‘₯βˆ’2 π‘₯<βˆ’1

{π‘₯+2 π‘₯β‰₯βˆ’1

ii) 𝑓(0)

A

𝑓(0) = 0 + 2 = 2

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7
Q

2. Sketch the following piecewise function

a) 𝑓(π‘₯)= {π‘₯βˆ’2 π‘₯<βˆ’1

{π‘₯+2 π‘₯β‰₯βˆ’1

iii) 𝑓(βˆ’1)

A

𝑓(βˆ’1) =βˆ’1 + 2 = 1

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8
Q

2. Sketch the following piecewise functions, stating the domain and range.

b) 𝑔(π‘₯)={2βˆ’π‘₯ π‘₯β‰€βˆ’1

{3 βˆ’1<π‘₯<3

{2π‘₯βˆ’5 π‘₯β‰₯3

A

b) 𝑔(π‘₯)={2βˆ’π‘₯ π‘₯β‰€βˆ’1

{3 βˆ’1<π‘₯<3

{2π‘₯βˆ’5 π‘₯β‰₯3

Domain: All real numbers OR βˆ’βˆž< x OR (βˆ’βˆž,∞)

Range:g(x)β‰₯1 OR [1,∞)

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9
Q

2. Sketch the following piecewise function

b) 𝑔(π‘₯)={2βˆ’π‘₯ π‘₯β‰€βˆ’1

{3 βˆ’1<π‘₯<3

{2π‘₯βˆ’5 π‘₯β‰₯3

i) 𝑔(βˆ’4)

A

b) 𝑔(π‘₯)={2βˆ’π‘₯ π‘₯β‰€βˆ’1

{3 βˆ’1<π‘₯<3

{2π‘₯βˆ’5 π‘₯β‰₯3

g(βˆ’4) = 2βˆ’βˆ’4 = 6

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10
Q

𝑦=|π‘₯|

A

with a vertical line on either side of the π‘₯.

|π‘₯|can be thought of as how far away from zero π‘₯ is, where direction is not important.

So no negatives. Hence |3|=3and |βˆ’3|=3as both +3and βˆ’3are both 3units away from the origin.

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11
Q

Graphing Absolute Functions
Looking at both graphs, where π‘₯ is positive (on the RHS of the y-axis), the graphs are
identical. But where π‘₯ is negative, (on LHS of 𝑦-axis) then 𝑦 = |π‘₯| is a mirror image of 𝑦 =
π‘₯ (reflected in the π‘₯-axis)

A

Domain and Range of 𝑦 = |π‘₯|
Clearly from the graph of 𝑦 = |π‘₯|

Domain: -∞ < π‘₯ < ∞
Range: 0 ≀ 𝑦 < ∞

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12
Q

Graph 𝑦 = |π‘₯ - 3| + 2

A
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13
Q

Sketch 𝑦 = -1 - |3π‘₯ + 2|

A
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14
Q

Hybrid or Piecewise Functions
Suppose you are recording some growth in population of some cells over time.

From 𝑑 = 0 to 𝑑 = 2 (hours)

the number of cells (in 100’s)

can be modelled by 𝐢(𝑑) = 𝑑2 + 1.

So at 𝑑 =0 we have

𝐢(0) = 02 + 1 = 1 (or 100) cells.

You observe that after 𝑑 = 2 the population is modelled by the equation 𝐢(𝑑) = 3𝑑 - 1 up until 𝑑 = 6
When we observe two or more functions over a set domain, we call it a hybrid or piecewise
function. Our example above can be written.

A
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15
Q
A
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16
Q
A
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17
Q

finding the Inverse Function

𝑓(π‘₯) = _1/2_π‘₯ + 5

A
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18
Q

Sketch

y=|x+ 2|

A
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19
Q

sketch

y=|x|βˆ’4

A
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20
Q

sketch

y=|x+ 3|βˆ’1

A
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21
Q

sketch

y=βˆ’|x|+ 2

A
22
Q

sketch

y=βˆ’|xβˆ’1|+ 3

A
23
Q

eval

f(-1)

A

f(βˆ’1)

=βˆ’(βˆ’1) + 4

= 1 + 4

= 5

24
Q

eval

f(0)

A

f(0)

=βˆ’0 + 4

= 4

25
Q

eval

f(1)

A

f(1)

= 2Γ—1βˆ’1

= 2βˆ’1

= 1

26
Q

eval

f(2)

A

f(2)

= 2Γ—2βˆ’1

= 4βˆ’1

= 3

27
Q

Sketch f(x)

A
28
Q

Determine the domain and range of f(x)

A

Domain: All (real) values of x

Range:f(x)β‰₯1

29
Q

eval

g(-3)

A

g(βˆ’3)

= 2βˆ’(βˆ’3)

= 2 + 3

= 5

30
Q

eval

g(-2)

A

g(βˆ’2)

= 2βˆ’(βˆ’2)

= 2 + 2

= 4

31
Q

eval

g(0)

A

g(0) = 3

32
Q

eval

g(3)

A

g(3)

= 2Γ—3βˆ’4

= 6βˆ’4

= 2

33
Q

Sketch g(x)

A
34
Q

Determine the domain and range of g(x).

A

Domain: All (real) values of x

Range: g(x)β‰₯2

35
Q

find its inverse function

f(x) = 7xβˆ’2

A
36
Q

find its inverse function

g(x) = 5e3x

A
37
Q

find its inverse function

f(x) =4x+3/5βˆ’1

A
38
Q

find its inverse function.

g(x) = 3/xβˆ’2

A
39
Q

find its inverse function.

h(x) = 3βˆ’2exβˆ’1

A
40
Q

find its inverse function

j(x) = 2 + 5 ln(x)

A
41
Q

Show that the inverse of

f(x) = x+ 1/xβˆ’1

is itself.

A
42
Q

Given the functions f(x) = 5x and g(x) = 2 cos(x), find the following composite functions.

f(g(x)) (This could also be written as fβ—¦g)

A

f(x) = 5x and g(x) = 2 cos(x)

f(g(x)) = f(2 cos(x))

5Γ—(2 cos(x))

= 10 cos(x)

43
Q

Given the functions f(x) = 5x and g(x) = 2 cos(x), find the following composite functions.

g(f(x)) (This could also be written as gβ—¦f)

A

g(f(x)) = g(5x)

= 2 cos(5x)

44
Q

Given the functions g(x) = 4βˆ’x2 and h(x) = 3xβˆ’2, find the following composite functions.

g(h(x)) (This could also be written as gβ—¦h)

A

g(x) = 4βˆ’x2 and h(x) = 3xβˆ’2

g(h(x)) = g(3xβˆ’2)

= 4βˆ’(3xβˆ’2)2

= 4βˆ’(9x2βˆ’12x+ 4)

= 4βˆ’9x2+ 12xβˆ’4

=βˆ’9x2+ 12x

45
Q

Given the functions g(x) = 4βˆ’x2 and h(x) = 3xβˆ’2, find the following composite function

h(g(x)) (This could also be written as hβ—¦g)

A

h(g(x)) = h(4βˆ’x2)

= 3(4βˆ’x2)βˆ’2

= 12βˆ’3x2βˆ’2

= βˆ’3x2+ 10

46
Q

Given the functions

f(x) = 4x+ 2,

g(x) = 1βˆ’3x

and h(x) = 2x2+ 1,

find the composite function fβ—¦gβ—¦h.

A
47
Q
A
48
Q

shown is old graph*

A
49
Q

State the (natural) domain and range for the function y=ex

make a table if it helps

A

Domain: All real values of x OR βˆ’βˆž< x <∞.

Range: All positive values of y OR 0< y <∞.

Note that y= 0 is NOT included in the range.

50
Q
A