7.3 Power Functions & 7.3 Power Functions & Function OperationsFunction Operations

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Operations on FunctionsOperations on Functions: for any : for any two functions f(x) & g(x)two functions f(x) & g(x)

1.1. AdditionAddition h(x) = f(x) + g(x)2.2. SubtractionSubtraction h(x) = f(x) – g(x)3.3. MultiplicationMultiplication h(x) = f(x)*g(x) OR f(x)g(x)4.4. DivisionDivision h(x) = f(x)/g(x) OR f(x) ÷ g(x)

5.5. CompositionComposition h(x) = f(g(x)) OR g(f(x))** DomainDomain – all real x-values that “make sense”

(i.e. can’t have a zero in the denominator, can’t take the even nth root of a negative number, etc.)

Ex: Let f(x)=3xEx: Let f(x)=3x1/31/3 & g(x)=2x & g(x)=2x1/31/3. Find (a) . Find (a) the sum, (b) the difference, and (c) the the sum, (b) the difference, and (c) the

domain for each.domain for each.

(a)3x1/3 + 2x1/3 = 5x1/3

(b)3x1/3 – 2x1/3 = x1/3

(c) Domain of (a) all real numbersDomain of (b) all real numbers

Ex: Let f(x)=4xEx: Let f(x)=4x1/31/3 & g(x)=x & g(x)=x1/21/2. Find (a) . Find (a) the product, (b) the quotient, and (c) the the product, (b) the quotient, and (c) the

domain for each.domain for each.(a) 4x1/3 * x1/2 = 4x1/3+1/2 = 4x5/6

(b)

= 4x1/3-1/2 = 4x-1/6 =

21

31

4

x

x

61

4

x

(c) Domain of (a) all reals ≥ 0, because you can’t take the 6th root of a negative number.

Domain of (b) all reals > 0, because you can’t take the 6th root of a negative number and you can’t have a denominator of zero.

564 x

6

4x

CompositionComposition• f(g(x)) means you take the function g and f(g(x)) means you take the function g and

plug it in for the x-values in the function f, plug it in for the x-values in the function f, then simplify.then simplify.

• g(f(x)) means you take the function f and g(f(x)) means you take the function f and plug it in for the x-values in the function g, plug it in for the x-values in the function g, then simplify.then simplify.

Ex: Let f(x)=2xEx: Let f(x)=2x-1-1 & g(x)=x & g(x)=x22-1. Find (a) -1. Find (a) f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the

domain of each.domain of each.(a) 2(x2-1)-1 =

122 x

(b) (2x-1)2-1

= 22x-2-1

= 142 x

(c) 2(2x-1)-1

= 2(2-1x)

=22x x

(d) Domain of (a) all reals except Domain of (a) all reals except x=x=±1.±1.

Domain of (b) all reals except x=0.Domain of (b) all reals except x=0.

Domain of (c) all reals except x=0, Domain of (c) all reals except x=0, because 2xbecause 2x-1-1 can’t have x=0. can’t have x=0.

Assignment