1. WARM UP: Linear Equations multiple choice Learning Targets :
TODAY IN CALCULUS…
WARM UP: Linear Equations multiple choice
Learning Targets :
You will be able to identify the domain and range of a function using its graph or equation.
You will be able to recognize even functions and odd functions using equations and graphs.
You will be able to interpret and find formulas for piecewise defined functions.
Independent practice

2. WARM UP:
Which of the following is an equation of the line through (−3, 4) with slope 1 2 ?
a) 𝑦−4= 1 2 (𝑥+3) b) 𝑦+3= 1 2 (𝑥−4)
c) 𝑦−4=−2(𝑥+3) d) 𝑦−4=2(𝑥+3)
2. Which of the following is an equation of the vertical line through
(−2, 4)?
a) 𝑦=4 b) 𝑥=2 c) 𝑦=−4 d) 𝑥=−2
3. Which of the following is the x-intercept of the line 𝑦=2𝑥−5?
a) 𝑥=−5 b) 𝑥=0 c) 𝑥= 5 2 d) 𝑥=− 5 2

3. 1.2 FUNCTIONS
FUNCTION: a relationship between two variables such that each independent value corresponds to exactly one dependent value
No repeated x values
Exactly one input paired with one output
Domain: independent values
Range: dependent values
FUNCTION NOTATION: 𝑦=𝑓(𝑥)
y is a function of x.
y is dependent on x.
EXAMPLE: 𝑃 𝑡 =−4.9 𝑡 2 −𝑡
P is a function of t.
P is dependent on t.
DOMAIN 𝒙
Independent variable
input
Function Machine
output
Dependent variable 𝒚
RANGE

4. EXAMPLE: Which of the equations below define y as a function of x?
1.2 FUNCTIONS
EXAMPLE: Which of the equations below define y as a function of x?
a. 𝑥+𝑦= b. 𝑥 2 + 𝑦 2 =1
c. 𝑥 2 +𝑦= c. 𝑥+ 𝑦 2 =1 X Y X Y
− 𝑥 − 𝑥
𝒚=𝟏−𝒙
− 𝑥 − 𝑥 2
𝑦 2 =1− 𝑥 2
𝒚=± 𝟏− 𝒙 𝟐 −1 1
1, −1 −2 −1 1 2 3 2 1 −1
FUNCTION
NOT A FUNCTION X Y −2 −1
3 , − 3
2 , − 2 1 X Y
− 𝑥 − 𝑥 2
𝒚=𝟏− 𝒙 𝟐
− 𝑥 − 𝑥
𝑦 2 =1−𝑥
𝒚=± 𝟏−𝒙 −2 −1 1 2 −3 1
FUNCTION
NOT A FUNCTION

5. 1.2 FUNCTIONS
PRACTICE: Which of the equations below define y as a function of x?
a. 𝑥−𝑦= b. 𝑥 2 + 𝑦 2 =4
c. 𝑥 2 −𝑦= c. 𝑥+ 𝑦 2 =2
− 𝑥 − 𝑥
−𝑦=−𝑥+1
− −1
𝒚=𝒙−𝟏
− 𝑥 − 𝑥 2
𝑦 2 =4− 𝑥 2
𝒚=± 𝟒− 𝒙 𝟐
NOT A FUNCTION
FUNCTION
− 𝑥 − 𝑥 2
−𝑦=− 𝑥 2
− −1
𝒚= 𝒙 𝟐
− 𝑥 − 𝑥
𝑦 2 =2−𝑥
𝒚=± 𝟐−𝒙
NOT A FUNCTION
FUNCTION

6. 1.2 FUNCTIONS: VERTICAL LINE TEST
PRACTICE: Identify which graphs are functions using the Vertical Line Test.
FUNCTION
FUNCTION
NOT A FUNCTION
NOT A FUNCTION
FUNCTION
FUNCTION

7. 1.2 DOMAIN AND RANGE
EXAMPLES: 3.
Function Machine
input
output 𝒙 𝒚
DOMAIN
RANGE
Independent variable
Dependent variable
−𝟐, ∞
[𝟎, ∞)
DOMAIN:
RANGE:
DOMAIN:
RANGE:
[−𝟐,𝟐]
[𝟎,𝟐]
DOMAIN: where the graph exists on the x-axis.
RANGE: where the graph exists on the y-axis.
DOMAIN:
RANGE:
(−∞, ∞)
[−𝟏,𝟏]

8. 1.2 DOMAIN AND RANGE
EXAMPLE 1: Find the domain and range, then sketch the graph of the function.
1. 𝑓 𝑥 = 𝑥 2 − 𝑓 𝑥 =− −𝑥
3. 𝑓 𝑥 = 4 −𝑥 𝑓 𝑥 =1+ 1 𝑥 2
DOMAIN:
RANGE:
DOMAIN:
RANGE:
(−∞,∞)
[−𝟗, ∞)
(−∞,𝟎]
(−∞, 𝟎]
DOMAIN:
RANGE:
DOMAIN:
RANGE:
(−∞,𝟎]
[𝟎,∞)
(−∞,𝟎)∪(𝟎, ∞)
(𝟏, ∞)

9. 1.2 DOMAIN AND RANGE
EXAMPLE 2: Use a graphing utility to graph the function. Then determine the domain and range of the function.
1. 𝑓 𝑥 = 3𝑥+2 𝑥<0 2−𝑥 𝑥≥ 𝑓 𝑥 = 𝑥 2 1−𝑥
3. 𝑓 𝑥 =5 𝑥 3 +6 𝑥 2 − 𝑓 𝑥 = 9− 𝑥 2
DOMAIN:
RANGE:
DOMAIN:
RANGE:
(−∞,∞)
(−∞,𝟐]
(−∞, 𝟏)∪(𝟏, ∞)
−∞,−𝟒 ∪[𝟎, ∞)
DOMAIN:
RANGE:
DOMAIN:
RANGE:
(−∞,∞)
[−𝟑,𝟑]
[𝟎,𝟑]

10. 1.2 SYMMETRY-EVEN AND ODD FUNCTIONS
A function 𝑦=𝑓(𝑥) is an…
EVEN FUNCTION if
𝑓 −𝑥 =𝑓(𝑥)
ODD FUNCTION if
𝑓 −𝑥 =−𝑓(𝑥)
(−𝑥, 𝑦)
(𝑥, 𝑦)
(𝑥, 𝑦)
(−𝑥, −𝑦)
Symmetry: about the y-axis
Symmetry: about the origin

11. 1.2 SYMMETRY-EVEN AND ODD FUNCTIONS
EXAMPLE: Determine whether the function is even, odd, or neither.
1. 𝑓 𝑥 = 𝑥 2 − 𝑓 𝑥 = 𝑥 5 − 𝑥 3 −𝑥
3. 𝑓 𝑥 = 𝑥 2 −2𝑥− 𝑓 𝑥 =1+ 1 𝑥 2
EVEN
ODD
EVEN
NEITHER

12. 1.2 SYMMETRY-EVEN AND ODD FUNCTIONS
EXAMPLE 2: Determine whether the function is even, odd, or neither without graphing.
1. 𝑓 𝑥 = 𝑥 2 − 𝑓 𝑥 = 𝑥 5 − 𝑥 3 −𝑥
3. 𝑓 𝑥 = 𝑥 2 −2𝑥− 𝑓 𝑥 =1+ 1 𝑥 2
𝑓 −𝑥 = (−𝑥) 2 −1
𝑓 −𝑥 = 𝑥 2 −1
𝑓 −𝑥 =𝑓(𝑥)
𝑓 −𝑥 = (−𝑥) 5 − −𝑥 3 −(−𝑥)
𝑓 −𝑥 =− 𝑥 5 + 𝑥 3 +𝑥
𝑓 −𝑥 =−𝑓(𝑥)
EVEN
ODD
𝑓 −𝑥 = (−𝑥) 2 −2 −𝑥 −1
𝑓 −𝑥 = 𝑥 2 +2𝑥−1
𝑓 −𝑥 =1+ 1 (−𝑥) 2
𝑓 −𝑥 =1+ 1 𝑥 2
𝑓 −𝑥 =𝑓(𝑥)
NEITHER
EVEN

13. 1.2 PIECEWISE FUNCTIONS
Different functions defined in certain domains.
EXAMPLE: 𝑓 𝑥 = −𝑥, 𝑥<0 𝑥 2 , 0≤𝑥≤1 1, 𝑥>1

14. 1.2 SYMMETRY-EVEN AND ODD FUNCTIONS
EXAMPLE 2: Write a piecewise formula for the function.
𝑓 𝑥 =
−𝑥+2,
0<𝑥≤2
− 1 3 𝑥+ 5 3 ,
2<𝑥≤5
(2, 1)
2, 1 5, 0
𝑦=𝑚𝑥+𝑏
𝑚= −1 3
0=− 𝑏
0=− 5 3 +𝑏
𝑏= 5 3
𝒚=− 𝟏 𝟑 𝒙+ 𝟓 𝟑
0, 2 2, 0
𝑦=𝑚𝑥+𝑏
𝑚= −2 2
𝑏=2
𝒚=−𝒙+𝟐
=−1

15. HOMEWORK #2: Pg.19: QR: 1-11odd E: 1-19odd, 21-30, 31-35odd,
37-40, 44-45
If finished, work on other assignments:
HW #1: Pg.9: QR:1-9odd
E:1-23odd, 27-33odd, 37, 41, 53