3.6-2 Composing, Decomposing Functions

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3.6-2 Composing, Decomposing
Functions
• Recall from yesterday, we can make a
combination of functions through using the
basic operations
– +, -, /, x
• Now, we can compose a new function by
essentially “embedding” one function into
another
Composition
• Composition = the composition of a function f
and g, denoted as f o g , is defined as f(g(x))
– Read as “f of g,” or “fog”
– Not the same as g(f(x))
• Just like yesterday, we can evaluate as a point,
OR as a new function with a new defined
domain
• After we compose a function, we can then use
it evaluate new function values
• (f ○ g)(x) is the new function
• (f ○ g)(5) means to replace every x, in the
composed function, with a 5
• Example. If f(x) = 2x – 3, and g(x) = x + 5, find:
• A) (f ○ g)(6)
• B) (f ○ g)(x)
• Example. If f(x) = x2 + 2, and g(x) = x, find the
formulas and state the domains for:
• A) (f ○ g)(x)
• B) (g ○ f)(x)
Decomposition
• On the other hand, we can decompose =
break down a function into simpler/separate
functions
• Best to work “inside-out”
• Look at the argument, and determine what
the function is doing to the argument
– Portion inside the main function
• Example. Decompose the function f(x) = (x4 +
1)3 into two functions.
• What is inside the argument?
• What is on the outside?
• Example. Decompose the function f(x) = 3 3 x 2  1
into two functions. And, three functions.
•
•
•
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Assignment
Pg. 270
#31-65 odd
Just find the formulas for fog on 41-53.
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