Understanding Composite Functions and Their Composition in Mathematics
110 likes | 241 Vues
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.
Understanding Composite Functions and Their Composition in Mathematics
E N D
Presentation Transcript
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)))
