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Precalc – 1.x

Precalc – 1.x. Combinations of functions. People buy a lot of plastic bottles. Then they throw them out and buy some more. Americans go through about 8.9 billion pounds of plastic bottles every year.

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Precalc – 1.x

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  1. Precalc – 1.x Combinations of functions

  2. People buy a lot of plastic bottles. Then they throw them out and buy some more. Americans go through about 8.9 billion pounds of plastic bottles every year. • About 27% of the bottles are recycled. The rest end up in the landfill. The percentage of decomposition after x years can be modeled by this function: d(x) = -.0004x2 + 100 • You need to figure out: • how much plastic is left after 10, 100, 300, and xyears • how many years it takes for all the plastic from one year’s worth of bottles to decompose

  3. Compositions of functions • Sometimes an input value will go through two processes – two separate functions. • For example: • f(x) = x + 1 g(x) = x + 2 f g a b c x+1 x+2 0 1 3

  4. Simplifying compositions • Instead of just applying two separate functions, we can write one function that takes them both into account. • Notation: The composition of f and g: • f(g(x)) OR f g(x) g f a b c f g(x)

  5. Simplifying Compositions: Process • Write one function that represents taking the input x into the function f(x)=x+1 and then into the function g(x)=x+2. AKA g(f(x)). • Start with: g(f(x)) = g(f(x)) • Write f’s algebra where f is: g(f(x)) = g(x+1) • Plug f into g, where x was: g(f(x)) = (x+1) + 2 • Simplify: g(f(x)) = x + 3

  6. Try an example. • f(x) = x + 3 g(x) = 4 – x2 • Find: (f g)(x) (g f)(x) (g f)(x)

  7. Arithmetic Combinations • (f+g)(x) = f(x) + g(x) • (f – g)(x) = f(x) – g(x) • (fg)(x) = f(x)g(x) • (f/g)(x) = f(x) / g(x), g(x) ≠ 0

  8. Examples • f(x) = 2x + 4 g(x) = x2 + x • Find these values • (f+g)(2) • (g/f)(8) • (gf)(1) • (fg)(1) • (g – f)(0)

  9. Examples • h(x) = x + 3 n(x) = x2 • Find: • (hn)(9) • (hn)(5) • (hn)(0) • (hn)(x) • (h/n)(9) • (h/n)(5) • (h/n)(0) • (h/n)(x)

  10. You have two email addresses, one through gmail and one through hotmail (which you’re phasing out).x: clubs you belong tog(x): the number of emails in your gmail inbox. g(x) = 3x + 5h(x): number of emails in your hotmail inbox. h(x) = x + 20How many emails will you have in total if you belong to…a) 1 club b) 3 clubs c) x clubs

  11. Come up with your own word problem that would require a PRODUCT or QUOTIENT combination of functions.

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