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5.1.1 Combining Functions - Part I Two functions: x 1 2 f(x) 2 6 g(x) 3 0 3 4 -1 New function, which we call f + g: x 1 2 3 (f+g)(x) 5 6 3 inputs (domain) f + g: outputs: Definition: same as f and g add outputs of f and g (f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x) (f . g)(x) = f(x) . g(x) (f / g)(x) = f(x) / g(x) if either f or g is undefined for some number, any combined function must be undefined for that number if g(a) = 0 for some value a, then (f /g)(a) is undefined (why?). Example: let f(x) = 2x + 3 and g(x) = x2 - 4 (f + g)(x) = f(x) + g(x) = (2x + 3) + (x2 - 4) = x2 + 2x - 1 5.1.1-1 Composition of Functions A new operation on functions, with a new operation symbol: o Notation: Pronunciation: Definition: fog "f composed with g" (f o g)(x) = f(g(x)) “f of g of x” Example: f(x) = 2x + 1, g(x) = x2 (f o g)(x) = f(g(x)) = f(x2) = 2x2 + 1 so: (f o g)(x) = 2x2 + 1 (Note that this is no more than brute-force substitution!!) and: (f o g)(3) = 2(32) + 1 = 19 ________________________________________________ Getting there quicker Suppose: (1) you don't need the general formula for a composition i. e. (f o g)(x) = 2x2 + 1 (2) you just want to compute one output: e.g. (f o g)(3) Do it the easy way! Don’t bother to compute the general formula first! Just do: (f o g)(3) = f(g(3)) = f(9) = 19 5.1.1-2