4-4 Functions Fogs, gofs and inverses

What is a function? A set of (x,y) pairs where _______________ ___________________________________ Functions can be _____________________ ___________________________________

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 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.

If and Then________________________ Now, what would g ◦ f be?

Domain of a composite The 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 ________________________________.

Example Find the domain of f(g(x) 1._____________________ 2._______________________________ 3. __________

1. If and find f ◦ g

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).

Inverse Inverse – ________________________ ________________________________ Notation for an inverse

How do I FIND an inverse? The easiest way is this: ______________ __________________________________ How to test? _______________________ __________________________________

Examples

Example: Show that F(x) and g(x) are inverses. Someone do it on the board!!

5-1 Exponential Rules Again?

Review of the Basics

5-2 Logarithms What is a logarithm?

Please graph y = 2 x By plotting points

Here it is! “logarithm” is the term for the inverse of any exponential function. y = 2 x  _____________________ This is written officially as __________

How to look at logarithms log b x = y can be thought of as log b n = p Then rewrite as n = b p (notice the significance of the variables chosen!!) And then solve it!

Some Rules 1._____________________ 2._____________________ 3._____________________ 4._____________________ 5._____________________ 6._____________________ 7. ______________________

1.Log Log 3

3.log b 16 = 2 4. log b =

5. log 6 x = -1 6.

5-3 Laws of Logarithms How to simplify equations so to solve.

There are 3 Laws of Simplification

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.

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!!

Class Work