4.3 Notes.notebook 1 October 25, 2012 4.3 Introduction to Composite Functions So far we have learned how to do the 4 arithmetic operations on functions. Now we switch to a very different operation. To look at composite functions we look at the idea of placing one function INSIDE another function. The inside function is evaluated first and its resulting value is placed inside the second function which is then evaluated. The key concept is to work from the INSIDE OUT.
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4.3 Notes.notebook
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October 25, 2012
4.3 Introduction to Composite Functions
So far we have learned how to do the 4 arithmetic operationson functions.
Now we switch to a very different operation.To look at composite functions we look at the idea of placingone function INSIDE another function. The inside functionis evaluated first and its resulting value is placed inside thesecond function which is then evaluated.
The key concept is to work from the INSIDE OUT.
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If f(x) = 2x 4 and g(x) = (x1)2
Then f(g(x)) means to substitute g(x) into the x of f(x).
Therefore f(g(x)) = 2(g(x))4 = 2(x1)2 4.This can then be evaluated into:
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This process can also work in the reverse direction.
Find g(f(x)). In other words substitue f(x) into the x of g(x).
Therefore g(f(x) = ((2x4) 1)2 This can then be evaluated into:
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Notice the second way to say f(g(x))
Use the above diagrams to find:
f(g(1)) __________
If f(g(x)) = 4 then solve for x: _____________
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Find the y value for g(1).This value then becomes the x value for f(x).
Now find g(f(4)) _________________
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Example 1:
above
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Example 2:
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Example 3:
c) f(f(3)) d) g(g(2))
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If you find g(1) and place this value into f,the result MUST be the same as h(1) ifh(x)=f(g(x))
In other words h(1) must equal f(g(1)). CHECK TO SEE.
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Let h(x)=f(g(x))Prove that h(1)=f(g(1))
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Homework:Page 298 #4,6,7,9, (10,11 Do not finddomain, range), 1215Multiple Choice #1,2Supplementary 2 Handout
Caution: Do not confuse the composition instruction(f o g)(x) with the multiplication instruction (f . g)(x)as used in some of the supplementary questions.
