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7.3 Power Functions & Function Operations. p. 415. Operations on Functions : for any two functions f(x) & g(x). Addition h(x) = f(x) + g(x) Subtraction h(x) = f(x) – g(x) Multiplication h(x) = f(x)*g(x) OR f(x)g(x) Division h(x) = f(x) / g(x) OR f(x) ÷ g(x)
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Operations on Functions: for any two functions f(x) & g(x) • Addition h(x) = f(x) + g(x) • Subtraction h(x) = f(x) – g(x) • Multiplication h(x) = f(x)*g(x) OR f(x)g(x) • Division h(x) = f(x)/g(x) OR f(x) ÷ g(x) • Composition h(x) = f(g(x)) OR g(f(x)) ** Domain – 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)=3x1/3 & g(x)=2x1/3. Find (a) the sum, (b) the difference, and (c) the domain for each. • 3x1/3 + 2x1/3 = 5x1/3 • 3x1/3 – 2x1/3 = x1/3 • Domain of (a) all real numbers Domain of (b) all real numbers
Ex: Let f(x)=4x1/3 & g(x)=x1/2. Find (a) the product, (b) the quotient, and (c) the domain for each. • 4x1/3 * x1/2 = 4x1/3+1/2 = 4x5/6 = 4x1/3-1/2 = 4x-1/6 = (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.
Composition • f(g(x)) means you take the function g and plug it in for the x-values in the function f, then simplify. • g(f(x)) means you take the function f and plug it in for the x-values in the function g, then simplify.
Ex: Let f(x)=2x-1 & g(x)=x2-1. Find (a) f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the domain of each. (a) 2(x2-1)-1 = (c) 2(2x-1)-1 = 2(2-1x) = (b) (2x-1)2-1 = 22x-2-1 = (d)Domain of (a) all reals except x=±1. Domain of (b) all reals except x=0. Domain of (c) all reals except x=0, because 2x-1 can’t have x=0.
