1.8 Combinations of Functions JMerrill, 2010

Arithmetic Combinations

Sum Let Find (f + g)(x)

Difference Let Find (f - g)(x)

Product Let Find

Quotient Let Find

You Do: Let Find: (f+g)(x) (fg)(x) (f-g)(x) (g-f)(x)

Finding the Domain of Quotients of Functions TTo find the domain of the quotient, first you must find the domain of each function. The domain of the quotient is the overlap of the domains.

Example TThe domain of f(x) = TThe domain of g(x) = [-2,2]

SSince the domains are: f(x) = g(x) = [-2,2] TThe domains of the quotients are

Composition of Functions MMost situations are not modeled by simple linear equations. Some are based on a system of functions, others are based on a composition of functions. AA composition of functions is when the output of one function depends on the input from another function.

Compositions Con’t  For example, the amount you pay on your income tax depends on the amount of adjusted gross income (on your Form 1040), which, in turn, depends on your annual earnings.

Composition Example IIn chemistry, the process to convert Fahrenheit temperatures to Kelvin units TThis 2-step process that uses the output of the first function as the input of the second function. This formula gives the Celsius temp. that corresponds to the Fahrenheit temp. This formula converts the Celsius temp. to Kelvins

Composition Notation  (f o g)(x) means f(g(x))  (g o f)(x) means g(f(x)

Composition of Functions: A Graphing Approach

You Do  f(g(0)) =  g(f(0)) =  (f°g)(3) =  (f°g)(-3) =  (g°f)(4) =  (f°g)(4) = f(x) g(x)

Compositions: Algebraically GGiven f(x) = 3x 2 and g(x) = 5x+1 FFind f(g(2))Find g(f(4)) gg(2)=5(2)+1 = 11 ff(11) = 3(11) 2 ==363 How much is f(4)? g(48) = 5(48)+1=241

Compositions: Algebraically Con’t GGiven f(x) = 3x 2 and g(x) = 5x+1 FFind f(g(x))Find g(f(x)) WWhat does g(x)=? ff(5x+1) ==3(5x+1) 2 ==3(25x 2 +10x+1) ==75x 2 +30x+3 What does f(x)=? g(3x 2 ) = 5(3x 2 )+1=15x 2 +1

You Do  f(x)=4x 2 -1g(x) = 3x  Find: