Week 8

Question 1

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

a) 𝑦=|𝑥+4|−1

start with 𝑦=𝑥+4

Answer 1

Question 2

1. Graph the following showing all steps in the transformation from the starting graph.
   b) 𝑦=3−|𝑥−2|

start with 𝑦=𝑥−2

Answer 2

(b) We have been asked to draw a graph of y= 3−|x−2|.

• Start with a graph of y=|x|(blue)
• Reflect across the x-axis: y=−|x|(green)
• Move the graph 2 to the right: y=−|x−2|(red)
• Move the graph up 3: y=−|x−2|+ 3 (black)

Question 3

1. Graph the following showing all steps in the transformation from the starting graph.
   c) 𝑦+2+|𝑥+1|=0

start with 𝑦=𝑥+1

Answer 3

(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=−|x|(green)
• Move the graph 1 to the left: y=−|x+ 1|(red)
• Move the graph down 2: y=−|x+ 1|−2 (black)

Question 4

2. Sketch the following piecewise functions, stating the domain and range.
   a) 𝑓(𝑥)= {𝑥−2 𝑥<−1

{𝑥+2 𝑥≥−1

Answer 4

a) 𝑓(𝑥)= {𝑥−2 𝑥<−1

{𝑥+2 𝑥≥−1

Domain: All real numbers OR −∞< x OR (−∞,∞)

Range:f(x)OR (−∞,−3) ∪ [1,∞)

Question 5

2. Sketch the following piecewise function

a) 𝑓(𝑥)= {𝑥−2 x <−1

{𝑥+2 𝑥≥−1

i) 𝑓(−3)

Answer 5

𝑓(−3) =−3−2 =−5

Question 6

2. Sketch the following piecewise function

a) 𝑓(𝑥)= {𝑥−2 𝑥<−1

{𝑥+2 𝑥≥−1

ii) 𝑓(0)

Answer 6

𝑓(0) = 0 + 2 = 2

Question 7

2. Sketch the following piecewise function

a) 𝑓(𝑥)= {𝑥−2 𝑥<−1

{𝑥+2 𝑥≥−1

iii) 𝑓(−1)

Answer 7

𝑓(−1) =−1 + 2 = 1

Question 8

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

b) 𝑔(𝑥)={2−𝑥 𝑥≤−1

{3 −1<𝑥<3

{2𝑥−5 𝑥≥3

Answer 8

b) 𝑔(𝑥)={2−𝑥 𝑥≤−1

{3 −1<𝑥<3

{2𝑥−5 𝑥≥3

Domain: All real numbers OR −∞< x OR (−∞,∞)

Range:g(x)≥1 OR [1,∞)

Question 9

2. Sketch the following piecewise function

b) 𝑔(𝑥)={2−𝑥 𝑥≤−1

{3 −1<𝑥<3

{2𝑥−5 𝑥≥3

i) 𝑔(−4)

Answer 9

b) 𝑔(𝑥)={2−𝑥 𝑥≤−1

{3 −1<𝑥<3

{2𝑥−5 𝑥≥3

g(−4) = 2−−4 = 6

Question 10

𝑦=|𝑥|

Answer 10

with a vertical line on either side of the 𝑥.

|𝑥|can be thought of as how far away from zero 𝑥 is, where direction is not important.

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

Question 11

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

Answer 11

Domain and Range of 𝑦 = |𝑥|
Clearly from the graph of 𝑦 = |𝑥|

Domain: -∞ < 𝑥 < ∞
Range: 0 ≤ 𝑦 < ∞

Question 12

Graph 𝑦 = |𝑥 - 3| + 2

Answer 12

Question 13

Sketch 𝑦 = -1 - |3𝑥 + 2|

Answer 13

Question 14

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

From 𝑡 = 0 to 𝑡 = 2 (hours)

the number of cells (in 100’s)

can be modelled by 𝐶(𝑡) = 𝑡² + 1.

So at 𝑡 =0 we have

𝐶(0) = 0² + 1 = 1 (or 100) cells.

You observe that after 𝑡 = 2 the population is modelled by the equation 𝐶(𝑡) = 3𝑡 - 1 up until 𝑡 = 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.

Answer 14

Question 15

Answer 15

Question 16

Answer 16

Question 17

finding the Inverse Function

𝑓(𝑥) = _1/2_𝑥 + 5

Answer 17

Question 18

Sketch

y=|x+ 2|

Answer 18

Question 19

sketch

y=|x|−4

Answer 19

Question 20

sketch

y=|x+ 3|−1

Answer 20

Question 21

sketch

y=−|x|+ 2

Answer 21

Question 22

sketch

y=−|x−1|+ 3

Answer 22

Question 23

eval

f(-1)

Answer 23

f(−1)

=−(−1) + 4

= 1 + 4

= 5

Question 24

eval

f(0)

Answer 24

f(0)

=−0 + 4

= 4

Question 25

eval

f(1)

Answer 25

f(1)

= 2×1−1

= 2−1

= 1

Question 26

eval

f(2)

Answer 26

f(2)

= 2×2−1

= 4−1

= 3

Question 27

Sketch f(x)

Answer 27

Question 28

Determine the domain and range of f(x)

Answer 28

Domain: All (real) values of x

Range:f(x)≥1

Question 29

eval

g(-3)

Answer 29

g(−3)

= 2−(−3)

= 2 + 3

= 5

Question 30

eval

g(-2)

Answer 30

g(−2)

= 2−(−2)

= 2 + 2

= 4

Question 31

eval

g(0)

Answer 31

g(0) = 3

Question 32

eval

g(3)

Answer 32

g(3)

= 2×3−4

= 6−4

= 2

Question 33

Sketch g(x)

Answer 33

Question 34

Determine the domain and range of g(x).

Answer 34

Domain: All (real) values of x

Range: g(x)≥2

Question 35

find its inverse function

f(x) = 7x−2

Answer 35

Question 36

find its inverse function

g(x) = 5e^3x

Answer 36

Question 37

find its inverse function

f(x) =4x+3/5−1

Answer 37

Question 38

find its inverse function.

g(x) = 3/x−2

Answer 38

Question 39

find its inverse function.

h(x) = 3−2e^x−1

Answer 39

Question 40

find its inverse function

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

Answer 40

Question 41

Show that the inverse of

f(x) = x+ 1/x−1

is itself.

Answer 41

Question 42

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 f◦g)

Answer 42

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

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

5×(2 cos(x))

= 10 cos(x)

Question 43

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 g◦f)

Answer 43

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

= 2 cos(5x)

Question 44

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

g(h(x)) (This could also be written as g◦h)

Answer 44

g(x) = 4−x² and h(x) = 3x−2

g(h(x)) = g(3x−2)

= 4−(3x−2)²

= 4−(9x²−12x+ 4)

= 4−9x²+ 12x−4

=−9x²+ 12x

Question 45

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

h(g(x)) (This could also be written as h◦g)

Answer 45

h(g(x)) = h(4−x²)

= 3(4−x²)−2

= 12−3x²−2

= −3x²+ 10

Question 46

Given the functions

f(x) = 4x+ 2,

g(x) = 1−3x

and h(x) = 2x²+ 1,

find the composite function f◦g◦h.

Answer 46

Question 47

Answer 47

Question 48

shown is old graph*

Answer 48

Question 49

State the (natural) domain and range for the function y=e^x

make a table if it helps

Answer 49

Domain: All real values of x OR −∞< x <∞.

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

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