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In this section, we explore the concept of composite functions and how to compose them to model various scenarios. A pit crew for a race car driver uses distance traveled as a function of time, d(t), and the amount of fuel left as a function of distance, f(d), to create a formula for fuel remaining with respect to time, denoted as (f ∘ d)(t). Additionally, we provide examples of calculating composite functions, writing their formulas, and decomposing complex functions into simpler components. This foundational knowledge is essential for deeper mathematical analysis.
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SECTION 4.1 • COMPOSITE FUNCTIONS
COMPOSITION: • A pit crew for a race car driver had a formula for the distance the car had traveled as a function of time: • d(t)
And they had a formula for the amount of fuel left in the car as a function of the distance traveled: f(d)
By hooking the two functions together in an operation called composition, the pit crew can obtain a formula for the amount of fuel remaining in the car as a function of time: • (f d) (t) or f ( d ( t ) )
EXAMPLE: • Let f(x ) = x 2 - 1 and g(x) = 1/x • Calculate (f g)(2) and (g f)(2) f(g(2)) f(1/2) (1/2)2 - 1 -3/4 g(f(2)) g(3) 1/3
f(x ) = x 2 - 1 and g(x) = 1/x • Write formulas for (f g) (x) and (g f) (x) and determine their domains. g(f(x)) f(g(x)) ( 1/x )2 - 1 1/x2 - 1
COMPOSITIONS AND THEIR DOMAINS DO EXAMPLES 3 AND 4
WHEN TWO COMPOSITIONS ARE EQUAL DO EXAMPLE 5
DECOMPOSING FUNCTIONS Let F(x) = (x 4 - 2) 3 Decompose F(x) as two functions. Let f(x) = x 4 - 2 and g(x) = x 3 Then F(x) = (g f) (x)
F(x) = (x 4 - 2) 3 Decompose F(x) as three functions. Let f(x) = x 3 , g(x) = x - 2 , h(x) = x 4 Then F(x) = f(g(h(x)))
