Interval Notation: (description) (diagram)

Open Interval: (1, 5) is interpreted as 1 < x < 5 where the

endpoints are NOT included.
(1, 5)
(While this notation resembles an ordered pair, in this context it
refers to the interval upon which you are working.)

Closed Interval: [1, 5] is interpreted as 1 < x < 5 where the

[1, 5]
endpoints are included.

Half-Open Interval: (1, 5] is interpreted as 1 < x < 5 where 1 is

(1, 5]
not included, but 5 is included.

Half-Open Interval: [1, 5) is interpreted as 1 < x < 5 where 1 is

[1, 5)
included, but 5 is not included.

Non-ending Interval: (1, ∞) is interpreted as x > 1 where 1 is

not included and infinity is always expressed as being "open"
(not included).

Non-ending Interval: (−∞, 5] is interpreted as x < -5 where 5 is

included and again, infinity is always expressed as being "open"
(not included).

Write the following descriptions using interval notation. Draw a number line if you need to.

1. 𝑥 ≤ 2 2. 𝑥 > −9 3. −4 ≤ 𝑥 ≤ −1 4. −3 ≤ 𝑥

5. 10 > 𝑥 6. 0 < 𝑥 ≤ 8 7. 𝑥 > −1 𝑎𝑛𝑑 𝑥 < 1

8. 𝑥 > 6 𝑜𝑟 𝑥 < −7 9. All real numbers 10. All real numbers except for 5

11. All real numbers except for 2 and -2 12. All even integer
1
All answers must be in interval notation, if possible.
Find the domain of the following relations and determine if it is a function.
1. 2. 3. 4.

D: D: D: D:
Function? Function? Function? Functi
on?

Find the range of the following relations.

5. 6. 7. 8.

Range: Range: Range: Range:

Composition of Functions
NOT multiplication!
( f  g)(x) Means find (f(g(x)), therefore g(x) will be your input for f(x))
ORDER MATTERS!

Let f(x) = x², g(x) = x – 1, h(x) = 2x + 5, j(x) = 6x, k(x) = x² + 2x + 1, and m(x) as shown in table. Find:

1)
( j  h)(12 ) 2) g(k(4)) 3) (h  f )(−5) 4) (m  h)(−2)

5) k(h(–1)) 6) g(j(h(4)) 7) ( f  m)(3)

2
8) (h  g )(x) 9) ( f  h )(x) 10) ( j  h)(x) 11) (k  j )(x)

12) (h  j )(x) 13) ( f  g )(x) 14) (h  h )(x) 15) (g  f )(x)

Day 1 HW: Functions & Composite Functions

I. Find f(-2) for the following functions.

___________1. f(x) = 3x + 1 ____________2. f(x) = 3x2 __________3. 𝑥 − |𝑥|

x −5
2
___________4. f(x) = x ___________5. f(f(-2)), f(x) = x2 – 1 _______6. f(x) = 3x2 + 5x + 3

II. Give the range of each function with the given domain.

____________________7. g(x) = 5 – 2x, D = {-1,0,1} ____________________8. h(x) = 2x2 + 1, D = {0,1,2}

____________________9. d(x) = x3, D = {–2,–1,0,1,2} ____________________10. j(x) = |x – 3| , D = {–5,–3,–1,1}

III. Given four functions t, s, f, and g, where:

t = {(7,5), (4,-6), (-3, -2), (5,7)} f(x) = 2x s = {(5,-6), (-6,5), (4,-3) (-3,8)} g(x) = x + 3
Find each value if it exists.

11. s(t(4)) _______ 12. t(s(4)) _______ 13. t(t(7)) ________ 14. s(s(4)) _______ 15. g(f(4)) _______

16. f(g(4)) _______ 17. g(f(0)) _______ 18. g(g(10)) _______ 19. s(f(-3)) _______ 20. f(g(x)) ________

IV. Given f(x) = 5 – 3x and g(x) = x2 + 2, find each of the following:

  1 
f f   
21. g(f(-3)) ________ 22. f(g(0)) ___________ 23.   3   ____________

24. f(g(a)) _________ 25. g(g(-a)) ___________ 26. g(f(x + h)) ___________

V. Given f(x) = 3x –5 , g(x) = x2 + 2, h(x) = x2 + 4x, and t(x) = –2x + 3

27. (f  g )(x) ____________ 28. (g  f)(x) _____________

29. (t  h)(x) ____________ 30. (h  t)(x) _____________

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 x +6
VI. Given f(x) = 2x – 6 , g(x) = 5x + 2, h(x) = 4 – x, and r(x) = 2

31. g(r(c+d)) ________________________ 32. f(g(h(a))) ________________ 33. h  g  f ______________________

Find the domain and range of the following relations and determine if it is a function.
1) 2) 3) 4)

D: D:
D: D:
R: R:
R: R:
Function? Function?
Function? Function?

5) 6) 7) 8)

*Periodic* D: D: D:
D:
R: R: R:
R:
Function? Function? Function?
Function?

Day 2 Warm up:

Let f(x) = 3x; g(x) = x2 + 4; h(x) = 2x – 3; and k(x) and j(x) be defined by the charts shown.

x k(x) x j(x)
-4 8 -2 1 1) ( f  g)(5) 2) (h  g)(x) 3) j (k (2)) 4) f (h(x ))
-2 2 0 -4
0 3 1 12
2 0 2 0
5) (g  f )(6) 6) k (k (2)) 7) (h  h)(x) 8) f (g( j (1)))

9) (h  g)(−2)

10) (g  h)(x) 11) h(g( f (x ))) ( ( ))

−1
12) k h 2

Answers -4 2x² + 5 3 4x−9 4x²−12x+13 6x−9 8 16 18x²+5 87 328 444 13

4
 1
13: Given: 𝑟(𝑥) = 3𝑥 + 1 𝑎𝑛𝑑 𝑠(𝑥) = 𝑥 − 3. Answer the following: Find the following:
2

a) r(s(0))= b) (r∘s)(-4)= c) (s∘r)(-1)= d) r(r(-2))= e) (r∘s)(16)= f) s(r(-4))=

14. Given: f(x) = -x2 + 8 and g(x) = √𝑥 + 5

a) 𝑓(𝑔(−1)) = b) (𝑓 ∘ 𝑔)(−3) = c) (𝑔 ∘ 𝑓)(−3) = d) 𝑓(𝑓(1)) = e) (𝑓 ∘ 𝑔)(11) =

Day 2 CW: Solving Absolute Value Equations

Absolute Value: the distance a number is from ___________ on a number line.
Describe each solution intuitively.
Example 1: |x| = 4 Example 2: |x + 2| = 3 Example 3: |x – 7| = 4

Solve and write the solution in set or interval notation. Check for extraneous solutions!
1) |2𝑥 − 1| + 7 = 14 2) −2|𝑥 + 3| − 1 = 5 3) −2|7 − 3𝑦| − 6 = −14

4) |𝑥 + 6| = 2𝑥 5) |𝑥 + 1| + 1 = 2𝑥 6) |𝑥 + 5| = |2𝑥 − 1|

7) |𝑥 2 − 3𝑥| = 4𝑥 − 12 8) |𝑥 − 1| = 𝑥 2 + 4𝑥 − 5

9) Write an absolute value equation with the given solutions. A) 4, -2 B) 4, 10

Solving Absolute Value Inequalities |𝑥| < 4 versus |𝑥| > 4

10) |𝑥 − 4| ≥ 0 11) |2𝑥 − 1| + 4 < 4 12) −5 + |𝑥 + 1| ≤ −3

13) 6 ≤ |𝑥 − 2| 14) |𝑥 − 2| < 2𝑥 − 7 15) |2𝑥| > 𝑥 − 1

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16) |2 − 𝑥| < 8 17) 3|4𝑥 − 1| ≤ 9 18) |𝑥 + 6| > 0

Write each compound inequality as an absolute value inequality.

19) 19 < 𝑥 < 37 20) −2.7 ≤ 𝑎 ≤ 5.5
Write an absolute value inequality and a compound inequality for each length x with the given tolerance:
18) a length of 4.5 m with a tolerance of 3 cm 19) a length of 4 ft with a tolerance of 6 inches

20) Fountain pens must weigh between 8 g and 12 g (inclusive) in order to pass quality inspection. Write an absolute
value inequality for the weight w of the fountain pen.

Day 2 HW: Absolute Value Equations. Solve each equation. Check for extraneous solutions.
1) |4𝑥| = 28 2) |3𝑥 + 6| = −12 3) |𝑧 − 1| = 7𝑧 − 13

4) |𝑠 + 12| = 15 5) |−3𝑥| = 63 6) 2|5𝑥 + 3| = 16

7) |6𝑥 + 7| = 5𝑥 + 2 8) |7𝑟 − 4| = 24 9) |3𝑐| + 2 = 11

10) 5|𝑥 + 1| + 6 = 21 11) |3𝑥 + 5| − 2𝑥 = 3𝑥 + 4 12) −|𝑑 + 2| = 7

13) |3𝑥 − 2| = |𝑥 − 6| 14) |𝑥 − 4| = |4 − 𝑥| 15) |𝑥 2 + 9𝑥 + 14| = 0

6
16) |𝑥 2 − 4| = −4 17) |𝑥 2 + 1| = 2𝑥 18) |𝑥 2 − 2𝑥| = 4𝑥 − 8

Write each compound inequality as an absolute value inequality:

1) 6.3 ≤ ℎ ≤ 10.3 2) −2.5 ≤ 𝑎 ≤ 2.5 3) 22 < 𝑥 < 33
Solve each inequality. Write answers in interval notation.
4) |𝑥 + 5| > 12 5) |𝑘 − 3| ≤ 19 6) |𝑥 + 2| ≥ 0

7) 2|𝑡 − 5| < 14 8) |3𝑥 − 2| + 7 ≥ 11 9) 5|2𝑏 + 1| − 3 ≤ 7

10) |2 − 3𝑤| ≥ 4 11) −3|7𝑚 − 8| < 5 12) |2𝑎| > 6

Write an absolute value inequality and a compound inequality for each length x with the given tolerance:
13) a length of 4.2 cm with a tolerance of 0.01 cm 14) a length of 10 ft with a tolerance of 1 in.

15) Write an absolute value inequality and a compound inequality for the temperature T that was recorded to be as low
as 65⁰F and as high as 87⁰F on a certain day.

16) The weight of a 40-lb bag of fertilizer varies as much as 4 oz from the stated weight. Write an absolute value
inequality for the weight w of a bag of fertilizer.

17) The duration of a telephone call to a software company’s help desk is at least 2.5 minutes and at most 25 minutes.
Write an absolute value inequality and a compound inequality for the duration d of a telephone call.

Review: Find the values/expressions for each function.

1. f(x) = 1 – 2x a) f(3) b) f(-5) c) f( a + 1) d) f( 1 – x)

1
2. g(x) = x2 + 3 a) g(14) b) 𝑔 ( ) c) g (x – 4) d) g( - 5)
3

7
 1
3. h(x) = 9 – x2 a) h(0) b) ℎ ( ) c) h( a + 3) d) h( -8)
2

4. f(x) = x2 – 1, a) f[g(2)] b) g[f(-3)] c) f[g(x)] d) f(a +1) – f(a)

g(x) = 1 – 2x

5. f(x) = x2 , a) f ° g b) g ° f
g(x) = x – 3

Day 3 warm up: Find the domain/range of each graph in interval notation. Tell whether it is a function.
1. 2. 3.

● °

Day 5 notes Piecewise Functions

Piecewise functions: functions that have pieces
Graph lines, fit to domain

3x + 2 if x  3

Ex.1 f(x) =  x − 1 if x  3

a) f(2) = _________

b) f(3) = _________

c) f(4) = _________

8
 x + 3 if x  −3

− 2
 x − 5 if x  −3
Ex.2 f(x) =  3

a) f(9) = _________

b) f(-3) = _________

c) f(-8) = _________

Ex. 3 f(x) =
x + 2 if x  −1

 1 1
− 2 x − 2 if x  −1

a. f(-2) = _________

b. f(-1) = _________

c. f(0) = __________

d. f(1) = __________

Ex. 4. Write a piecewise function for the given graph to the right:

Evaluate the given function for the given value of x.

1
 x − 10, if x  6
h (x ) =  2
− x − 1, if x  6
 17. h(1) ______ 18. h(10) _______ 19. h(0) ________ 20. h(6) ________

Graph the function on GRAPH PAPER.

2x + 13, if x  −5
2x if x  1  3x − 14, if x  4
f (x ) =  1
f (x ) =  f (x ) = 
x + , if x  −5
21. − x + 3, if x  1 22.  2 23. − 2x + 6, if x  4

9
 − 1, if 0  x  1

3, if − 1  x  2 − 3, if 1  x  2
 
5, if 2  x  4 f (x ) = − 5, if 2  x  3
f (x ) =  − 7, if 3  x  4
8, if 4  x  9 
10, if 9  x  12 − 9, if 4  x  5
24.  25.

Day 5 HW
1. 2. 3.

a) f(-1) a) g(0) a) h(-1)

b) f(0) b) g(3) b) h(4)

c) f(1) c) g(6) c) h(7)

Graph the following piecewise functions, (#4,5 have the given functions already graphed, bold according to domain)

4. 5. 6.

 x +3 if -4 ≤ x < -1

y = 4 if x = -1
x
7. 1)  if x > -1

Domain: ________________________

Range: ________________________

10
Evaluate the function for the given value of x. 𝑓(𝑥) = {5𝑥 − 1, 𝑓𝑜𝑟 𝑥 < −2 − 𝑥 − 9, 𝑓𝑜𝑟 − 2 ≤ 𝑥 < 3 2√𝑥 − 3 +
1
1, 𝑓𝑜𝑟 𝑥 ≥ 3 𝑔(𝑥) = { 𝑥 2 − 10, 𝑓𝑜𝑟 𝑥 < 6 − |𝑥 − 1|, 𝑓𝑜𝑟 𝑥 ≥ 6
2

8. f(-4) – g(6) 9. 2f(-2) + 3g(0) 10. f(7) – g(-10) 11. f(15)

Day 5 HW: Writing Piecewise Functions

Directions: Write an equation for each of the following piecewise functions.
1. 2.

𝑓(𝑥) = { 𝑓(𝑥) = {

𝑓(𝑥) = { 𝑓(𝑥) = {
3. 4.

5.
𝑓(𝑥) = {

6. Evaluate 2f(-3) + 3f(-6) – f(4)

if 𝑓(𝑥) = {−2𝑥 − 7 𝑓𝑜𝑟 𝑥 < −5 − (𝑥 + 2)2 + 6 𝑓𝑜𝑟 − 5 ≤ 𝑥 < 0 √𝑥 − 1 𝑓𝑜𝑟 𝑥 ≥ 0

7. Find 2h(-4) + 3h(2) +4g(-1) – g(-3)

if ℎ(𝑥) = {−2𝑥 𝑓𝑜𝑟 𝑥 < 1 3𝑥 + 5 𝑓𝑜𝑟 𝑥 ≥ 1 𝑔(𝑥) = {𝑥 2 + 4 𝑓𝑜𝑟 𝑥 < −2 𝑥 3 𝑓𝑜𝑟 𝑥 ≥ −2

11
Day 7 CW:
Absolute Value Function (Desmos Demonstration)
https://www.desmos.com/calculator/rarxiatpip

Parent Function: _______________

Vertex: _______________________

Domain: ______________________

Range: _______________________

Increasing: _______________________

Decreasing: ________________________
Transformations/Graphing 𝑦 = 𝑎|𝑥 − ℎ| + 𝑘 Vertex (h, k)
Order of Transformations: From left to right: a,h,k
a: vertical stretch or compression h: horizontal shift left or right k: vertical shift up or down
-a: reflection over the x-axis
Graph each equation or inequality. Describe the transformations done to the parent graph 𝑦 = |𝑥| IN ORDER.
1) 𝑦 = 2|𝑥 − 4| 2) 𝑦 = 3|𝑥 + 2| − 2 3) 𝑦 = −2|𝑥 − 2| + 4

1
4) 𝑦 = −3|𝑥| + 3 5) 𝑓(𝑥) < 2|𝑥 − 3| − 4 6) 𝑓(𝑥) > 2 |𝑥 − 3|

12
 Graph the given function then find the value(s) of x for which:
(a) 𝑓(𝑥) = 0 (b) 𝑓(𝑥) > 0 (Positive) (c) 𝑓(𝑥) < 0 (Negative)
1
7) 𝑓(𝑥) = 2|𝑥 + 3| − 4 8) 𝑓(𝑥) = |𝑥 − 3| 9) 𝑓(𝑥) = −3|𝑥 + 2| − 1
2

(a) (a) (a)

(b) (b) (b)

c) c) c)

Increasing: Increasing: Increasing:

Decreasing: Decreasing: Decreasing:

Day 7 HW
Graph the function on GRAPH PAPER. Identify the vertex, and describe the transformations compared to y = |x|.
1
1. y = 3|x| 2. f(x) = -3|x| 3
3. f(x) = |x| 4. y = |x – 2| 5. y = |x| - 2
1
6. y = |x + 2| 7. y = -|x + 9| + 3 8. y = 2 |x – 8| -1 9. y = 2|x| - 7 10. y = -2|x + 5|

Write an equation of the graph shown.

11. 12.
13.

13
Review 1
State the domain and range. Is the relation a function?
1. {(-3,2), (-2,1), (-1,0), (-1, 1), (0,1)}

2. 3. 4.

5. Use the graph in #2

a. Write the equation for the graph .
b. Find the x and y intercept(s).
c. Interval(s) of increase:
d. Interval(s) of decrease:
e. Interval(s) where the graph is positive:
f. Interval(s) where the graph is negative:

Given f(x), find the following.

6. f(-3) if f(x) = 10x + 3x2 7. f(x – 2) if f(x) = 5x2 – 8x

Graph on graph paper.

8. g(x) = |2x+4| 9. f(x) = - |x| + 3 10. y = |3x| 11. -2x + 7  |y|

3x + 2, if x  3

12. Given: f(x) =  x − 1, if x  3 find 2f(0) – 3f(3) + 5f(6)
2 2
 x+ , if x  2
3 3

13. Graph on GRAPH PAPER: f(x) = − x + 1, if x  2

Identify the vertex, direction of opening, describe transformations to y = |x|, and intervals of increasing and decreasing.
1
14. y = 6|x – 7| 15. y = -|x + 8| -3 16. y = 3 |x| + 7

14
Solve and write the solutions in interval notation if possible.
1
1. |2x + 8| + 2 = 1 2. |x + 4| - 1 =6x 3. |3𝑥 + 6| − 2 = 2
3

4. |4 + x| = 1 – 2x 5. |2 – 3x| - 5x = 4 6. 8 ≤ |2𝑥 − 6|

7. |3𝑥 − 6| < 𝑥 − 4 8. |𝑥 − 7| = |2𝑥 − 2|

9. ℎ(6) 10. 𝑔(−5) 11. 𝑓(2) + 𝑓(−4) − 2𝑔(0) 12. 3ℎ(−4) + 5ℎ(−2) − ℎ(−1)

19. Graph: 𝑓(𝑥) = {1 − 2 ≤ 𝑥 < −1, |𝑥| − 2 − 1 < 𝑥 < 1, − |𝑥 − 2| + 2 1<𝑥≤3

Find Domain, Range, intervals of increase and decrease.

For problems 20-29, graph each piecewise function.

15
  x + 3 if x  −1 x −1 if x  3
f ( x) =  f ( x) = 
20. 2 x − 1 if x  −1 21. 2 if x  3

−1 if x  0 4 − x if x  2
f ( x) =  f ( x) = 
22. x − 3 if x  0 23. 3x − 6 if x  2


−2 x if x  −1
 2 + x if x  −2
f ( x) = 3x − 1 if − 1  x  2

 1 f ( x) = − x if − 2  x  1
− x if x  2 0 if x  1
24.  2 25. 

 x if x  −2 −2 if x  0
f ( x) =  f ( x) = 
26. 2 x if x  −2 27.  2 if x  0

2 x − 1 if x  0

2 x + 1 if x  −1 f ( x) = 2 − x if 0  x  3
f ( x) =  x +1
28. 2 x + 2 if x  −1 29.  if x  3

For problems 30-32, give the piecewise function that each graph represents.

30. 31. 32.

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