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Basic Properties, Inverse Functions, Composite Functions FUNCTIONS

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1. FUNCTION BASICS, NOTATION, DEFINITIONS

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BASIC PROPERTIES OF A FUNCTION

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TESTING FOR FUNCTIONS

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GRAPHS OF FUNCTIONS PRECISION IS KEY.

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TEST YOUR UNDERSTANDING

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HOW TO WRITE A FUNCTION

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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

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TEST YOUR UNDERSTANDING WRITTEN IN INTERVAL NOTATION

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TEST YOUR UNDERSTANDING

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2. COMPOSITE FUNCTIONS

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PLUGGING ONE FUNCTION INTO ANOTHER

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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)

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TEST YOUR UNDERSTANDING

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CHECK YOUR UNDERSTANDING

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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))?

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3. INVERSE FUNCTIONS

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IMPORTANT!!!! This is a reciprocal function: This can also be written as f(x) = x -1 This is NOT an inverse function!

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AN INVERSE FUNCTION…

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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)=

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INVERSE FUNCTION - EXAMPLE

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IMPORTANT

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

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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!

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4. OTHER FEATURES AND BASIC CONCEPTS

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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.

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EXAMPLES OF SIGN DIAGRAMS

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IMPORTANT

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CHECK YOUR UNDERSTANDING

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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.

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CHECK YOUR UNDERSTANDING

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REFERENCE Contents provided by Mathematics for the international student Mathematics SL 2 nd Edition Hasse and Harris Publications, 2010

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