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6.3 Power Functions & Function Operations What is the difference between a power function and a polynomial equation? What operations can be performed on functions? What is a composition of two functions? How is a composition of functions evaluated?

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Operations on Functions: for any two functions f(x) & g(x) 1.A ddition h(x) = f(x) + g(x) 2.S ubtraction h(x) = f(x) – g(x) 3.M ultiplication h(x) = f(x)*g(x) OR f(x)g(x) 4.D ivision h(x) = f(x) / g(x) OR f(x) ÷ g(x) 5.C omposition h(x) = f(g(x)) OR g(f(x)) ** D omain – 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.)

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Let f(x)=3x 1/3 & g(x)=2x 1/3. Find (a) f(x) + g(x) (b) f(x) – g(x) (c) the domain for each. Let f(x)=3x 1/3 & g(x)=2x 1/3. Find (a) f(x) + g(x) (b) f(x) – g(x) (c) the domain for each. (a)3x 1/3 + 2x 1/3 = 5x 1/3 (b)3x 1/3 – 2x 1/3 = x 1/3 (c)Domain of (a) all real numbers Domain of (b) all real numbers

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. Find (a) the product, (b) the quotient, and (c) the domain for each. Let f(x)=4x 1/3 & g(x)=x 1/2. Find (a) the product, (b) the quotient, and (c) the domain for each. (a)4x 1/3 * x 1/2 = 4x 1/3+1/2 = 4x 5/6 (b) = 4x 1/3-1/2 = 4x -1/6 = (c) Domain of (a) all reals ≥ 0, because you can’t take the 6 th root of a negative number. Domain of (b) all reals > 0, because you can’t take the 6 th root of a negative number and you can’t have a denominator of zero.

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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.

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Let f(x)=2x -1 & g(x)=x 2 -1. Find (a) f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the domain of each. (a) 2(x 2 -1) -1 = (b) (2x -1 ) 2 -1 = 2 2 x -2 -1 = (c) 2(2x -1 ) -1 = 2(2 -1 x) = (d) D DD 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.

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What is the difference between a power function and a polynomial equation? The power tells you what kind of equation—linear, quadratic, cubic… What operations can be performed on functions? Add, subtract, multiply, divide. What is a composition of two functions? A equation (function) is substituted in for the x in another equation (function). How is a composition of functions evaluated? Write the outside function and substitute the other function for x.

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