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Basic Properties, Inverse Functions, Composite Functions FUNCTIONS.

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Presentation on theme: "Basic Properties, Inverse Functions, Composite Functions FUNCTIONS."— Presentation transcript:

1 Basic Properties, Inverse Functions, Composite Functions FUNCTIONS

2 1. FUNCTION BASICS, NOTATION, DEFINITIONS

3 BASIC PROPERTIES OF A FUNCTION

4 TESTING FOR FUNCTIONS

5 GRAPHS OF FUNCTIONS PRECISION IS KEY.

6 TEST YOUR UNDERSTANDING

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8 HOW TO WRITE A FUNCTION

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10 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

11 TEST YOUR UNDERSTANDING WRITTEN IN INTERVAL NOTATION

12 TEST YOUR UNDERSTANDING

13 2. COMPOSITE FUNCTIONS

14 PLUGGING ONE FUNCTION INTO ANOTHER

15 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)

16 TEST YOUR UNDERSTANDING

17 CHECK YOUR UNDERSTANDING

18 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))?

19 3. INVERSE FUNCTIONS

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

21 AN INVERSE FUNCTION…

22 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)=

23 INVERSE FUNCTION - EXAMPLE

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25 IMPORTANT

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

27 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!

28 4. OTHER FEATURES AND BASIC CONCEPTS

29 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.

30 EXAMPLES OF SIGN DIAGRAMS

31 IMPORTANT

32 CHECK YOUR UNDERSTANDING

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38 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.

39 CHECK YOUR UNDERSTANDING

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


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