Do Now from 1.3a: Which of the Twelve Basic Functions:
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Do Now from 1.3a: Which of the Twelve Basic Functions:Do Now from 1.3a: Which of the Twelve Basic Functions:
• Do Do not not have a domain of all real numbers?have a domain of all real numbers?
• Have a domain of all real numbers Have a domain of all real numbers exceptexcept zero? zero?
• Have no negative numbers in their domain? Of theseHave no negative numbers in their domain? Of these two, which one is defined at zero?two, which one is defined at zero?
1f x
x f x x lnf x x
1f x
x
f x x lnf x x
Which of the Twelve Basic Functions:Which of the Twelve Basic Functions:
1f x
x intf x x
f x x
sinf x x
2f x x cosf x x
• Have points of discontinuity? Are these points in theHave points of discontinuity? Are these points in the domain of the function?domain of the function?
• Are bounded (above Are bounded (above and and below)?below)?
• Are even?Are even?
ContinuousContinuousFunctionFunction
Not aNot aContinuousContinuous
FunctionFunction
cosf x x 1
1 xf x
e
Now, on to more practiceNow, on to more practicewith Section 1.3b:with Section 1.3b:
Today, we analyzeToday, we analyzefunctions graphicallyfunctions graphically
and explore piecewiseand explore piecewisefunctions…functions…
Quality Practice Problems
Graph the given function, then answer the following questions:
22y x 1. On what intervals is the function increasing or decreasing?
2. Is the func. even, odd, or neither?
3. What are the extrema of the func.?
4. Does the graph relate to one of the 12 basic functions? If so, how?
Inc: [ 2, )Inc: [ 2, )88
Dec: (– , 2 ]Dec: (– , 2 ]88
NeitherNeither
Min. of 0 at Min. of 0 at xx = 2 = 2
The squaring func.,The squaring func.,shifted right 2shifted right 2
5. What is the domain and range of the function?
D: (– , )D: (– , )88 88R: [ 0, )R: [ 0, )88
Quality Practice Problems
Graph the given function, then answer the following questions:
4h x x 1. On what intervals is the function increasing or decreasing?
2. Is the func. even, odd, or neither?
3. What are the extrema of the func.?
4. Does the graph relate to one of the 12 basic functions? If so, how?
Dec: [ – 4, )Dec: [ – 4, )88
NeitherNeither
Max. of 0 at Max. of 0 at xx = – 4 = – 4
The square root func.,The square root func.,shifted left 4, reflectedshifted left 4, reflectedacross across xx-axis-axis
5. What is the domain and range of the function? D: [ – 4, )D: [ – 4, )88 R: ( – , 0 ]R: ( – , 0 ]88
Quality Practice Problems
Graph the given function, then answer the following questions:
5 abs 1g x x
1. On what intervals is the function increasing or decreasing?
2. Is the func. even, odd, or neither?
3. What are the extrema of the func.?
4. Does the graph relate to one of the 12 basic functions? If so, how?
NeitherNeither
Max. of 5 at Max. of 5 at xx = –1 = –1
The abs. val. function,The abs. val. function,reflected across reflected across xx-axis,-axis,shifted left 1, up 5shifted left 1, up 5
5. What is the domain and range of the function? D: (– , )D: (– , )88 88 R: (– , 5 ]R: (– , 5 ]
Inc: ( – , –1 ]Inc: ( – , –1 ]88
Dec: [ –1, )Dec: [ –1, )88
88
So, what are these functionsSo, what are these functionswith “smart parts?”with “smart parts?”
Piecewise Functions!!!Piecewise Functions!!!
Which of the twelve basic functions has the followingpiecewise definition over separate intervals of its domain?
f(x) =x if x > 0
–x if x < 0
This is the absolute value function!!!This is the absolute value function!!!
Sketch the given function (without using your calculator!), listany points of discontinuity, and state the domain and range ofthe function.
2 , 0
, 0
x xf x
x x
Function is continuous!Function is continuous!
D: (– , )D: (– , )88 88
R: [0, )R: [0, )88
Sketch the given function (without using your calculator!), listany points of discontinuity, and state the domain and range ofthe function.
3 , 0
, 0x
x xg x
e x
Point of discontinuityPoint of discontinuityat at xx = 0 = 0
D: (– , )D: (– , )88 88
R: (– , 0 ] U ( 1, )R: (– , 0 ] U ( 1, )88 88
Sketch the given function (without using your calculator!), listany points of discontinuity, and state the domain and range ofthe function.
1 , 0
, 0
x xw x
x x
Point of discontinuityPoint of discontinuityat at xx = 0 = 0
D: (– , )D: (– , )88 88
R: (– , )R: (– , )88 88
Sketch the given function (without using your calculator!), listany points of discontinuity, and state the domain and range ofthe function.
2
, 0
, 0
x xf x
x x
Function is continuous!Function is continuous!
D: (– , )D: (– , )88 88
R: [ 0, )R: [ 0, )88
Sketch the given function (without using your calculator!), listany points of discontinuity, and state the domain and range ofthe function.
2 , 1
, 1 1
int , 1
x x
f x x x
x x
Point of discontinuityPoint of discontinuityat at xx = 2, 3, 4, 5,… = 2, 3, 4, 5,…
D: (– , )D: (– , )88 88
R: [ 0, )R: [ 0, )88
Whiteboard Problems
Graph the given function, then answer the following questions:
3r x x 1. On what intervals is the function increasing or decreasing?
2. Is the func. even, odd, or neither?
3. What are the extrema of the func.?
4. Does the graph relate to one of the 12 basic functions? If so, how?
EvenEven
Min. of 3 atMin. of 3 at x x = 0= 0
The abs. val. function,The abs. val. function,shifted up 3shifted up 3
5. What is the domain and range of the function?
D: (– , )D: (– , )88 88R: [ 3, )R: [ 3, )
Inc: [ 0, )Inc: [ 0, )88
Dec: (– , 0 ]Dec: (– , 0 ]8888
Whiteboard Problems
Graph the given function, then answer the following questions:
2xf x e
1. On what intervals is the function increasing or decreasing?
2. Is the func. even, odd, or neither?
3. What are the extrema of the func.?
4. Does the graph relate to one of the 12 basic functions? If so, how?
Inc: (– , )Inc: (– , )88
NeitherNeither
No extremaNo extrema
The exp. function,The exp. function,shifted up 2shifted up 2
5. What is the domain and range of the function?
D: (– , )D: (– , )88 88R: ( 2, )R: ( 2, )
8888
Homework: p. 110 29-51 odd
Remember your WP that is due on Monday, and Monday is also the quiz for 1.1 – 1.3