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Combining Functions |
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When we write the rules f(x)= 2x2 - 3x + 5 we are changing functions by multiplying by constants and adding functions together (x2 is multiplied by 2, x is multiplied by -3, 2x2, -3x and 5 are all added together). The functions we call polynomials are formed this way. We can add or subtract functions (e.g., s(x) = x - 2 + (2x + 3)2 or d(x) = x - 2 - (2x + 3)2), multiply functions ( p(x) = (x - 2)(2x + 3)2), divide functions (q(x) = (x - 2)/(2x + 3)2) ). If f(x) = x - 2 and g(x) = (2x + 3)22, then s = f + g, d = f - g, p = fg and q = f/g (f*g)(x) = f(x) * g(x), where * is + or - or multiplication or division) In the exercises and examples you have seen several that ask you to write a rule in the form h(-x), h(2 - x), Quad(1/x). In each of these cases we are combining two functions (e.g., h and the negative-x function or h and the x-> 2-x, or Quad and the reciprocal function). Of course, we could combine these in the opposite order. Would we get the same result? Suppose h(x) = x2 + 1/x + 2 and Quad(a) = 100 + 21a - a2. What are the results if we reverse the order that created h(-x). How would you write the new function in function notation? What is its rule? Now reverse the order that created Quad(1/x). How would you write this new function in function notation? What is its rule? We define this new way of combining functions to be the function f o g or g o f where (f o g)(x) = f(g(x)) and (g o f)(x) = g(f(x)] ; (f o g) needs to have g defined and to have g(x) in the domain of f
If f(x) = ((7x - 3) / 2)1.3 + 1/(7x - 3) , write f as the composition of two functions. What is the domain of the function f? A weather balloon is inflated in such a way that its radius increases at the rate of 4 cm/sec. Express the volume of the balloon as a function of time and determine the volume of the balloon after 4 seconds. (volume of sphere is 4Pi(r3)/3). |
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