Composite Functions - Notes

Example 19 building and testing a composite function

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Example 1.9: Building and Testing a Composite Function f (x) = x 2 – 4x g (x) = x + 2 What is f (g (x))? To find the composite, plug g (x) into f (x), just as you would with any number. f (g (x)) = (x + 2) 2 – 4 (x + 2) Then simplify. f (g (x)) = (x 2 + 4x + 4) – 4x – 8) f (g (x)) = x 2 – 4 Let’s test it. f (g (x)) means do g, then f . What happens if we start with x = 7? 7 g (x) 7 + 2 = 9 f (x) (9) 2 4(9) = 45 So, if it worked, our composite function should do all of that in one step. 7 x 2 – 4 = (7) 2 – 4 = 45 It worked! There is a different notation that is sometimes used for composite functions. Your textbook will use both. f (g (x)) which very naturally conveys the idea of “plugging g (x) into f (x).” and f ○ g (x), which more naturally conveys the idea of “doing one function, and then the other, in sequence.” The two notations mean the same thing. Remember you can also have g (f (x)) which means the same as g ○ f (x).
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