1.5 Algebra of Functions

1.5 Algebra of Functions

Given 2 functions fandg, we can put them together and make new functions • Sum (f + g)(x) = f (x) + g(x) • Difference (f – g)(x) = f (x) – g(x) • Product (f •g)(x) = f (x) • g(x) • Quotient • Domain of the combo function is those values each have in common

Ex 1) Find the new function and state the domain • f + g • f – g • f •g • d) = (f + g)(x) = f (x) + g(x) D = {x: x ≥ 1} = (f – g)(x) = f (x) – g(x) D = {x: x ≥ 1} = (f •g)(x) = f (x) • g(x) D = {x: x ≥ 1} D = {x: x > 1}

We can combine functions in real world situations too! Ex 2) The efficiency of an engine with a given heat output, in calories, can be calculated by finding the ratio of two fncts of heat input, D and N, where D(I) = I – 5700 & N(I) = I. a) Write a function for the efficiency of the engine in terms of heat output. b) Find efficiency when heat input is 17,200 cal

Another way to combine functions is called a composition of functions. Given two functions f and g, the composite function, f ◦ g is (read f of g of x) Domain is the set of elements of g such that g(x) is in domain of f *Domains of f ◦ g and g ◦ f are not necessarily the same (work from the inside out)

Ex 3) Find composition and state domain of each a) b) D = {x: x≠ 0}

We can even “decompose” a function into two functions Ex 4) Determine two functions f and g such that Can you see a function inside another function? inside, this is g so then

Homework #105 Pg 36 #1-11 odd, 8, 13, 15, 23, 27, 29, 31, 33, 34, 37, 41

1.5 Algebra of Functions