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SECTION 4.1

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).

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SECTION 4.1

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  1. SECTION 4.1 • COMPOSITE FUNCTIONS

  2. 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)

  3. And they had a formula for the amount of fuel left in the car as a function of the distance traveled: f(d)

  4. 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 ) )

  5. 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

  6. 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

  7. COMPOSITIONS AND THEIR DOMAINS DO EXAMPLES 3 AND 4

  8. WHEN TWO COMPOSITIONS ARE EQUAL DO EXAMPLE 5

  9. 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)

  10. 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)))

  11. CONCLUSION OF SECTION 4.1

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