Basic Properties, Inverse Functions, Composite Functions FUNCTIONS

Basic Properties, Inverse Functions, Composite Functions

FUNCTIONS

1. FUNCTION BASICS, NOTATION, DEFINITIONS

BASIC PROPERTIES OF A FUNCTION

TESTING FOR FUNCTIONS

GRAPHS OF FUNCTIONS• PRECISION IS KEY.

TEST YOUR UNDERSTANDING

TEST YOUR UNDERSTANDING

HOW TO WRITE A FUNCTION

HOW TO WRITE A FUNCTION

DOMAIN AND RANGE• DOMAIN – LIST OF ALL X VALUES

• RANGE – LIST OF ALL Y VALUES ONE WAY YOU OFTEN SEE THIS ON THE IB:

• INTERVAL NOTATION

• WRITE THE BEGINNING AND ENDING VALUES OF THE INTERVAL

• [ OR ] IMPLIES THAT THE VALUE IS INCLUDED IN THE INTERVAL

• ( OR ) IMPLIES THAT THE VALUE IS NOT INCLUDED

• INFINITY IS ALWAYS WRITTEN WITH A ROUND BRACKET

TEST YOUR UNDERSTANDING• WRITTEN IN INTERVAL NOTATION

TEST YOUR UNDERSTANDING

2. COMPOSITE FUNCTIONS

PLUGGING ONE FUNCTION INTO ANOTHER

IN LAYMAN’S TERMS• f(g(x)) implies that you take the equation for g(x) and insert it into f(x) for everywhere you

see an x.

• Another way to write this is (f o g)(x)

TEST YOUR UNDERSTANDING

CHECK YOUR UNDERSTANDING

TO THE NEXT LEVEL• Assume f(x) = 5x-2

• Assume g(x) = x-10

• What do you think f(g(x)) is?

• What do you thing g(f(x)) is?

• I challenge you to find an instance where f(g(x)) = g(f(x))

• What do you think g(f(3)) is? How would you solve that?

• What about f(g(-2))?

3. INVERSE FUNCTIONS

IMPORTANT!!!!• This is a reciprocal function:

• This can also be written as f(x) = x -1

• This is NOT an inverse function!

AN INVERSE FUNCTION…

HOW TO WRITE AN INVERSE OF A FUNCTION

• TO RE-WRITE A FUNCTION AS THE INVERSE, FOLLOW THESE SIMPLE STEPS:• RE-WRITE f(X)= AS y=• SWITCH x AND y• SOLVE FOR y• REWRITE y= AS f-1(x)=

INVERSE FUNCTION - EXAMPLE

IMPORTANT

IMPORTANT

The graph of a function and it’s inverse are symmetrical about the line y=x

INVERSE FUNCTIONS AND COMPOSITES

One way to determine if two functions are inverses of one another is to use composite functions. It is known that:

This is one way to check if you correctly found the inverse of a function.

This is also one way to determine if two seemingly unrelated functions are inverses of one another.

Try a few simple examples on your own!

4. OTHER FEATURES AND BASIC CONCEPTS

SIGN DIAGRAMS• A quick representation of the graph’s behavior.

• Will come in handy as we begin to study calculus next year.

• Can help with trying to figure out the appearance of a graph of a complicated function.

EXAMPLES OF SIGN DIAGRAMS

IMPORTANT

CHECK YOUR UNDERSTANDING

CHECK YOUR UNDERSTANDING

CHECK YOUR UNDERSTANDING

CHECK YOUR UNDERSTANDING

CHECK YOUR UNDERSTANDING

CHECK YOUR UNDERSTANDING

CAUTION• The next example (rational functions) is going to be covered in a few days. If you don’t

completely ‘get it’ no worries! We’ll cover it completely soon.

CHECK YOUR UNDERSTANDING

CHECK YOUR UNDERSTANDING

REFERENCE• Contents provided by Mathematics for the international student Mathematics SL 2nd

Edition

• Hasse and Harris Publications, 2010