This document discusses restrictions on the domains of composite functions. It notes that if the domains of the original functions f(x) and g(x) are all-inclusive, then the domain of the composite function f(g(x)) will also be all-inclusive. However, if the domains of f(x) or g(x) are restricted, then the domain of the composite function will also be restricted. It provides examples of finding the domains and ranges of composite functions based on the original functions.

1. 4.4 Notes.notebook February 21, 2013
4.4 Restrictions on Composite Functions
If the domain of 2 functions is
then the domain of their composition will be the same
and have no restrictions.
Ex) Look at f(x)=2x­1 g(x)=x2­1 and f(g(x)).
Notice that the LEFT screen has been set so that only
the composite function is displayed.
What is the equation of f(g(x))? Notice its domain?
__________________
What is the range of f(x), of g(x)?
How would we find the range of f(g(x))?
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2. 4.4 Notes.notebook February 21, 2013
Use the same 2 equations to find the equation for g(g(x))
Again, notice its domain is ALL inclusive.
Use your calculator to find the range of this function.
Your screen should look like this:
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3. 4.4 Notes.notebook February 21, 2013
So far we conclude that if the domains of functions are
all inclusive, so is the domain of their composite.
The range is more difficult to determine and might require
the help of a calculator.
What if the domains of the original functions are NOT all
inclusive. What about the composite function's domain??
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4. 4.4 Notes.notebook February 21, 2013
If and g(x) = x+3
state the domain of both f(x) and g(x) and find the domain
of:
1) f(g(x)) Hint: Find the composite function and then determine
its domain.
2) g(f(x))
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5. 4.4 Notes.notebook February 21, 2013
Remember, domains are restricted by Non­Permissible Values.
Denominators must NOT equal 0.
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6. 4.4 Notes.notebook February 21, 2013
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7. 4.4 Notes.notebook February 21, 2013
In this case, compose the function and then remember that Square Root
functions only exist for positive values. This will determine the restrictions on
the domain.
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8. 4.4 Notes.notebook February 21, 2013
Now try these:
This one has issues! The square root disappears in
the composition, but since the domain of f(x) is restricted,
so is the domain of g(f(x)). Be careful.
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9. 4.4 Notes.notebook February 21, 2013
Now let's get creative.
The final type of question GIVES the composite function
and asks for the original functions that would produce this
composition. Obviously in many cases, more than one answer
will work.
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10. 4.4 Notes.notebook February 21, 2013
a) g(x) = _________ b) g(x) = ___________
f(x) = _________ f(x) = ____________
Does your f(g(x) work?
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11. 4.4 Notes.notebook February 21, 2013
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12. 4.4 Notes.notebook February 21, 2013
Homework: Page 314 #3, 4(use calc. for
range), 5(in class), #6a)b)c) [d for challenge]
7a)c), 9a)b), 10, 11a)b), 12, 14a) challenge
Mult. Choice #1,2.
Supplementary 3 Handout
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