4-4 Functions

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4-4 Functions

• Fogs, gofs and inverses

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What is a function?

• A set of (x,y) pairs where _______________
• ___________________________________
• Functions can be _____________________
• ___________________________________
• ___________________________________

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1. If and
find
c) f(x) g(x)
a) f(x) g(x)
b) f(x) g(x)
2. For the above functions find
a) f(3) g(3)
b) f(1)g(2)
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Fog? Gof? What is that?

• fog __________________________
• gof __________________________
• That notation indicates composition. That is,
  we are taking the composition of 2 (or more)
  functions.

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This is a composition

• can be thought of as the composition of the
  functions
• and
• You sort of tuck one function into the other.

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

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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
  ________________________________.

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Example Find the domain of f(g(x)

• _____________________
• _______________________________
• 3. __________

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4-5 Inverse Functions
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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? ____________
• ___________________________________
• ___________________________________
• ___________________________________

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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).

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Inverse

• Inverse ________________________
• ________________________________
• Notation for an inverse

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How do I FIND an inverse?

• The easiest way is this ______________
• __________________________________
• __________________________________
• How to test? _______________________
• __________________________________
• __________________________________

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Examples
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Group Problem Show that F(x) and g(x) are
inverses.
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5-1 Exponential Rules

• Again?

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Review of the Basics
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5-2 Logarithms

• What is a logarithm?

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Please graph y 2x
By plotting points
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Here it is!

• logarithm is the term for the inverse of any
  exponential function.
• y 2x ? _____________________
• This is written officially as __________

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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!

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Some Rules

• _____________________
• _____________________
• _____________________
• _____________________
• _____________________
• _____________________
• 7. ______________________

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• Log525
• 2. Log3

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• logb16 2
• 4. logb

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• 5. log6x -1
• 6.

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5-3 Laws of Logarithms

• How to simplify equations so to solve.

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There are 3 Laws of Simplification
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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.

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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 Id suggest 3)
• Now, just so you know, if there is no base
  written down, it means base 10. (yes, write that
  down!! Ill fool you with it if you arent
  careful!!

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