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