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.