precalc – 1.x
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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.
• 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 x years– how many years it takes for all the plastic from one
year’s worth of bottles to decompose
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
a f gb c
0 x+1 x+21 3
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)
a g fb c
f g(x)
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
Try an example.
• f(x) = x + 3 g(x) = 4 – x2
• Find: (f g)(x) (g f)(x) (g f)(x)
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
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)
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)
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 + 20
How many emails will you have in total if you belong to…a) 1 club b) 3 clubs c) x clubs
Come up with your own word problem that would require a
PRODUCT or QUOTIENT combination of functions.
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