Not a function, function, one-to-one?How to draw an inverse given sketch

Finding Inverses

Domain and Range of f, f -1

Domain of Composite Functions

Graph Exponential Functions

Verify it’s an inverse

Evaluating Composite Functions

Log Domains

Graph Log Functions

Solve Logs

Interest Rates

Exponential Growth

Terms of a and b, Evaluate Logs

Expand, Compress Logs

Exponential Decay\ Half-life

Please report any errors ASAP by email to sakim@fjuhsd.net or IM at kimtroymath.

Problems may be more difficult on test. Consult homework assignment. Not all topics covered.

Type of relation. Draw inverse

NOT A FUNCTION: fails vertical line test

FUNCTION: pass vertical line test only.

ONE-TO-ONE: passes vertical and horizontal line test.

Be able to explain why it is a function, one-to-one, or not a function.

Just like to find the inverse of coordinates we switch the x and y, we do the same with graphs.

( , )3 4( , )-1 2

( , )-2 0Remember, to draw freehand, it’s a reflection over the line y = x

Finding Inverses Switch the x and the y, solve for y.

Important, when you have to find inverses of rational expressions that have fractions, you will probably have to cross multiply, move the y’s to one side and FACTOR!

52

13)(

x

xxf

52

13

y

yx

13)52( yyx

1352 yxxy

1532 xyxy

15)32( xxy

32

15

x

xy

1)( 3 xxf

32

15)(1

x

xxf

13 yx31 yx

3 33 1 yx

yx 3 1

)(1 13 xfx

12)(

coming) isanswer (Only

one Try this

x

xxf

x

xor

x

xxf

21

12)(1

A couple of things to note, this was a case where f and f -1 happened to be the same, it doesn’t happen normally. The answers to the left are the same, it all just depends on which way you moved the terms.

Verify it’s an inverse.

32

15)(

52

13)(

f) ff f prove to(Choose

inverses are functions Verify the

1

1-1-

x

xxf

x

xxf

or

If it’s an inverse:

(f ◦ f -1)(x) = x

and

(f -1 ◦ f)(x) = x

You will probably only be asked to verify in one direction though.

(f ◦ f -1)(x) = x (f -1 ◦ f)(x) = x

Clear: (f ◦ f -1)(x) = x Clear: (f -1 ◦ f)(x) = x

53215

2

13215

3))(( 1

xxxx

xff

Multiply by the common denominator to get rid of fractions! Careful when distributing.32

32

x

x

)32(5152

)32(1153

xx

xx

1510210

32315

xx

xx

xx

17

17

Be careful when distributing, be careful with signs.

35213

2

15213

5))(( 1

xxxx

xff 52

52

x

x

)52(3132

)52(1135

xx

xx

15626

52515

xx

xx

xx

17

17

Multiply by the common denominator to get rid of fractions! Careful when distributing.

Be careful when distributing, be careful with signs.

,

2

3

2

3,

,2

5

2

5,

Domain and Range of f, f -1

32

15)(

52

13)( 1

x

xxf

x

xxf

Remember:

Domain of f = range of f -1

Range of f = domain of f -1

,2

5

2

5,

,

2

3

2

3,

:

:

fofRange

fofDomain

Look at the previous three slides, I used the same problems. You can probably expect to have to find the inverse, verify, and find the domain and range given a function.

3

2)(

f. of range and domain, inverse, theFind

x

xxf

x

x

x

xxf

2

3

2

3)(1

),2()2,(:

),3()3,(:

fofRange

fofDomain2

3

3)2(

32

23

3

2x

x

xy

xxy

xyxy

yxxy

y

yInverse Work

13)( 2 xxxf1

1)(

x

xh

))3((

)2)((

))5((

fh

hf

gf

215)5( g

)2(f 1)2(322 1))5((gf

2

112

1)2(

h

)1(f 1)1(3)1( 2 5))2(( hf

1

11)3(3)3()3( 2 f

)1(h11

1

2

1))3(( fh

1

1)( xxg

Evaluating Composite Functions

f(x)

g(x)

)2)(())3(( fggf 3 )(f1

1 )(g3

Domains of Composites

4

2)(

x

xf3

1)(

x

xg

))((

ofdomain theFind

xfg 4x

g(x)for 3 xhavet Can'

t workdon'that

values- xfind to

3f(x) Make 3

4

2

x

x

x

x

3

10

1232

)4(32

,44,

3

10

3

10,:D

1) First, find domain of f(x) [note, letters changed, same process.]

2) Find domain of g(x), and find values of f(x) that equal the excluded values.

3) Combine in interval notation.

coming)answer(only

))((

ofdomain theFind

xff

,

2

9

2

9,44,:D

Graph Exponential Functions

1) Factor

2) Transformations

1) Compress\Stretch

2) Reflect

3) Shift

3) Key Points transformation

1)

4) Asymptote

1) y = 0

5) Graph

aa

1,1;,1;1,0

132 42 xy

132 )2(2 xy

1 upShift

axis-Reflect x

2stretchVert

2leftShift2

1compressHoriz

Remember, reciprocal for horizontal compression stretch.

3

1,

2

55,

2

3)1,2(

)3

1,1()3,1()1,0(

a = 3

3

1,

2

55,

2

3)1,2(

3

2,

2

56,

2

3)2,2(

3

2,

2

56,

2

3)2,2(

3

1,

2

53,

2

3)1,2(

3

1,

2

13,

2

1)1,0(

)3

1,1()3,1()1,0(

1y

Asymptote follows vertical shift (up\down)

Point Work

Clear

Log Domains

Everything inside the log is greater than zero.

Check to see where it is zero AND undefined.

Make a number line!

)4log()( 2xxf

4

4log

2x

xy

04 2 x0)2)(2( xx

-2 2

x = -3 x = 0 x = 3

– –+

)2,2(: D

Open circle, there is no ‘equals to’

Pick Positive Region, want greater than zero.

REMEMBER: include the boundary points, use just use ( ), not [ ]

04

42

x

x

0)2)(2(

4

xx

x

-4 -2 2

x = -5

–

x = -3

+

x = 0

–

x = 3

+

),2()2,4(: D

Graph Log Functions

1) Factor

2) Transformations

1) Compress\Stretch

2) Reflect

3) Shift

3) Key Points transformation

1)

4) Asymptote

1) x = 0

5) Graph

1,

1;1,;0,1a

a

1)2

1(log2

2

1 xy

12

1log2

2

1

xy 1 upShift

2stretchVert

reflection axis-y

2stretchHoriz

Remember, reciprocal for horizontal compression stretch.

)1,4()3,1()1,2(

)1,2()1,2

1()0,1(

a = _1_

2

1,43,1)1,2(

2,42,1)0,2(

1,41,1)0,2(

1,41,1)0,2(

)1,2(1,2

1)0,1(

0x

Asymptote follows horizontal shift (left\right)

Point Work

Clear

Be warned, it is possible for both logarithmic and exponential you can have a fractional base. But you do the problem the same way. So don’t panic, we live in a beautiful world.

Terms of a and b, Evaluate logs.

a

MM

MrM

NMN

M

NMMN

raMa

a

b

ba

ar

a

aaa

aaa

ra

M

aa

a

log

loglog

loglog

logloglog

logloglog

log

1log01loglog

Terms of a and b problems Break down the log into factors

Use the log rules to break them down

Then substitute.

7ln2ln ba

28ln

72ln 2

7ln2ln 2 7ln2ln2

ba 2

3

14

1ln

3

1

72

1ln

72

1ln

3

1

72ln1ln3

1

))7ln(2(ln1ln3

1

))7ln(2(ln03

1

ba 3

1

Evaluating logs. Watch out for change of base formula. Sometimes, after you change the base, you might put it back together.

2log6log 444

3log36log 63

9log4log 66

36log9log 44

These are the main types. Look at the tricky one on pg 308: 21, 22 also.

The key to these problems is to know your rules and use order of operations properly. Remember, if there is an exponent, take care of the exponent first.

12log44 12

6log

3log

3log

36log

6log

36log

36log6

2

36log6

2

36

9log4

4

1log4

1

Solve Logs. Know your rules, don’t do anything illegal, and DOUBLE CHECK!!!! √√ Remember exact vs. approximate

a

MM

MrM

NMN

M

NMMN

raMa

a

b

ba

ar

a

aaa

aaa

ra

M

aa

a

log

loglog

loglog

logloglog

logloglog

log

1log01loglog

)2(log)4(log2 22 xx

2 logs, same base, try to make the insides equal.

)2(log)4(log 22

2 xx )2()4( 2 xx xxx 21682

016102 xx0)2)(8( xx

28 xx

4)12(log)5(log 33 xx

Logs with same base, number, use log rules to combine, then switch to exponential form. Remember to combine first.

4))12)(5((log3 xx43))12)(5(( xx

815112 2 xx076112 2 xx0)4)(192( xx

44

19

xx

321 34 xx

Exponential form, log both sides. You may have to FACTOR out x to help solve.

321 3ln4ln xx

3ln)32(4ln)1( xx3ln33ln24ln4ln xx

4ln3ln33ln24ln xx4ln3ln3)3ln24(ln x

)3ln24(ln

4ln3ln3

x

This is an exact answer, if it’s on the calculator part, you may be asked to approximate.

Also note, if the bases happen to be the same, you can just make the exponents equal to each other and solve.

If you see something in the form below, you should probably solve by factoring.

3)3(92 xx

3)3(32 2 xx

xu 3

032 2 uu0)1)(32( uu

12

3

uu

132

33

xx

0x

There are also other tricky examples such as Pg 313: 39 – 41, 45 – 49.

It’s important that you follow log rules. Many of you broke the log rules on the red problem on the test. Watch out for that.

Expand (sum and difference of logs) Compress Logs (single log term) WATCH OUT FOR FACTORING, PARENTHESIS

a

MM

MrM

NMN

M

NMMN

raMa

a

b

ba

ar

a

aaa

aaa

ra

M

aa

a

log

loglog

loglog

logloglog

logloglog

log

1log01loglog

10

12log1log4

2 xxx

10

12log1log

24 xx

x

12

101log

2

2

xxx

2

2

)1(

101log

xx

10log1

)ln(2

1ln

1ln 2 xx

x

x

)ln(2

11ln 2 xx

x

x

xx

x

x 2

1

2

11ln

)1(

1

2

11ln

xx

x

x

22

1ln

x

2

22 1

110log

x

x

22 1

110log2

x

x

22 1log)110log(2 xx

)1)(1(log2))1(10log(2 2

1

xxx

))1log()1(log(2)1log(10log2 2

1

xxx

))1log()1(log(2)1log(

2

110log2 xxx

))1log()1(log(2)1log(

2

112 xxx

)1log(4)1log(4)1log(2 xxx

Watch out for factoring, such as the blue green step. Be careful about how you split things apart.

Find the principal amount if the amount due after 4 years is $200 at 4% interest compounded continuously.

Interest Rates

tr

tn

PeA

n

rPA

trPI

Interest Compound Continuous

1

Interest Compound

Interest Simple

Be able to:

- Find amount due.

- Principal amount.

- Years invested.

If it says ‘compounded continuously’ you should probably use the continuous compound interest formula.

Suggestion: show set up incase you type it into your calculator incorrectly.

)25.2(52

52

032.1200

A

What is the amount due after 2.25 years if $200 is invested at 3.2% compounded weekly?

CAREFUL WITH CALCULATOR!!!!!

Compound Interest

P = 200 Principal

r = .032 Interest Rate (decimal)

n = 52 # of times compounded per year

t = 2.25 Length of investment

93.214A

)4(04.200 Pe

)174.1(200 P

P43.170

Continuous Compound Interest

A = 200 Amount Due

r = .04 Interest Rate (decimal)

t = 4 Length of investment

How long was the money invested for if the principal of $16000 returned $25000 at 4.75% interest compounded daily?

Compound Interest

P = 16000 Principal

A = 25000 Amount Due

r = .0475 Interest Rate (decimal)

n = 365 # of times compounded per year.

)(365

365

0475.11600025000

t

)(365

365

0475.1

16

25t

)(365000130137.1ln)5625.1ln( t

000130137.1ln)365()5625.1ln( t

t 40.9000130137.1ln365

)5625.1ln(

Also study effective interest rate. It may or may not show up.

Be careful with rounding.

Use approximate symbols where necessary.

Exponential Growth

0)( 0 keNtN kt

N0 represents initial number of cells.

k represents growth rate of cells.

Amount at t = 0.

Find growth rate.

Find population after time t.

Find how long it takes for something to reach a particular population.

How long does it take to double, triple, multiply be some amount.

Other various items.

There are 4 bacteria in a culture that follows unbounded exponential growth. After 3 days, there are 10.

What is the growth rate?

This means find ‘k’

3410 ke10 = N(3) Population after three days.

4 = No Initial Population

3 = t Time (in days)

3

2

5 ke

3ln2

5ln ke

k32

5ln

305.325

lnk

Find the population after 5 days

Round to the nearest whole number.

Plug in 5 for t)5(305.4)5( eN

4 = No Initial Population

5 = t Time (in days)

.305 = k Growth rate

18)5( NIt is possible for me to ask the second question without the first. You would need to know that you need to figure out k first.

(Different problem) How long until a population triples if the growth rate is 5% (units in days)

)(05.3 tNeN 3N is because you triple the amount N.

)(05.3 te)(05.ln3ln te

t05.3ln dayst 97.21

05.

3ln

Don’t be thrown off by the wording. It could be bacteria, people, stress molecules, who knows. Understand what the questions are asking and what you need to find.

Exponential Decay

0)( 0 keNtN kt

N0 represents initial number of cells.

k represents decay rate of cells.

Amount at t = 0.

Find decay rate.

Find population after time t.

Find how long it takes for something to reach a particular population.

How long does it take to cut in half, a third, some other amount.

Other various items.

Everything pretty much works the same as growth. The only different style problem involves half-life. I will show the long way to do the problem, and then the shortcut should be in the notes.

The half life of radioactive isotope Kim-302 is 12 years. Find the decay rate of the Kim isotope.

)12(

2

1 kNeN

)12(

2

1 ke

)12(ln2

1ln ke

k122

1ln

k12

21

ln