4-4 Functions

Fogs, gofs and inverses

What is a function?

A set of (x,y) pairs where _______________

___________________________________

Functions can be _____________________

___________________________________

___________________________________

2f(x) x 3

b) f(1)+g(2)

a) f(x) + g(x) b) f(x) · g(x) c) f(x) ÷ g(x)

a) f(3) · g(3)

1. If and find

1

g(x)x

2. For the above functions find

Fog? Gof? What is that?

“fog” __________________________

“gof” __________________________

That notation indicates “composition”. That is, we are taking the composition of 2 (or more) functions.

This is a composition

can be thought of as the composition of the functions

and

You sort of “tuck” one function into the other.

3x 3

If and

Then________________________

Now, what would g◦f be?

f(x) x 3g(x) x 3

Domain of a compositeThe domain of a composite is all x in the domain of

the inner function for which the values are in the domain of the outer function.

Just determine _____________________, then exclude anything that is not in the domain ________________________________.

2

( )

1( )

2

g x x

f xx

ExampleFind the domain of f(g(x)

1. _____________________

2. _______________________________

3. __________

x 1

g(x)x 5

1. If and find f ◦ g

x 6

f(x)x 4

4-5 Inverse Functions

What is an inverse function?

Two functions are said to be inverses if

___________________ That is, the composite of a function with its own inverse is x.

Now what does that mean? ____________

___________________________________

___________________________________

___________________________________

So, lets experiment

Let and

What is f(g(x))?

How about g(f(x))?

Find f(0), f(1), f(2) and f(3).

Now find g(0), g(1), g(4) and g(9).

2f(x) x g(x) x

Inverse

Inverse – ________________________

________________________________

Notation for an inverse

___________________

How do I FIND an inverse?

The easiest way is this: ______________

__________________________________

__________________________________

How to test? _______________________

__________________________________

__________________________________

Examples

21. Find the inverse of f(x) x 6

2. Find the inverse of f(x) x 3

x 2

3. Find the inverse of f(x)x 3

Example:

Show that F(x) and g(x) are inverses.

( ) 2 6

1( ) 3

2

f x x

g x x

Someone do it on the board!!

5-1 Exponential Rules

Again?

Review of the Basicsm n1. a a ___

m n2. a a ___

05. a _

n6. a ___

n7. a ___m n3. (a ) ___

n4. (ab) ___ m n8. b b ____

x 3 11. 2

4

x 32. 2 2

x 3 x3. 4 2 128

2x4. 8 16

x 35. 4 1

x x6. 3 3 18

x x x 17. 3 3 3

729

x x8. 6(4 4 ) 96

x x x x 19. 3 3 3 3 54

5-2 Logarithms

What is a logarithm?

Please graph y = 2x

By plotting points

Here it is!

“logarithm” is the term for the inverse of any exponential function.

y = 2x _____________________

This is written officially as __________

How to look at logarithms

logbx = y can be thought of as logbn = p

Then rewrite as n = bp (notice the significance of

the variables chosen!!)

And then solve it!

Some Rules

1. _____________________

2. _____________________

3. _____________________

4. _____________________

5. _____________________

6. _____________________

7. ______________________

1. Log525

2. Log3

13

x

x

3. logb16 = 2

4. logb = 32

18

5. log6x = -1

6. 2log 34

5-3 Laws of Logarithms

How to simplify equations so to solve.

There are 3 Laws of Simplification

b b1. log M log N ______

b b2. log M log N ______

pb3. log M ______

There is also a Rule

This is called the Change of Base Rule: It can be used to convert a problem so that you could solve it on the calculator. It also has extensive use in Calculus in derivatives and integration.

blog M ___________

How does it work, you ask?

First, solve this the normal way. Then, solve using

change of base rule (that is, pick a new base –

I’d suggest 3)

Now, just so you know, if there is no base written down, it

means base 10. (yes, write that down!! I’ll fool you

with it if you aren’t careful!!

9log 27 p

4 4 41. log x log 3 log 2

4 42. log x log 3 2

8 8

23. log (x 3) log x

3

4 4 74. log (x) log x log 7

Class Work

3 3 75. log x log 5 log 7

2 2 36. log (x 1) log (3x 1) log 243