Composition of Functions
Date: 07/23/99 at 03:00:47 From: Melanie Pichelmann Subject: Algebraic operations How do I solve: f(x) = x+2 and g(x) = 3x-1, then f(g(x)) = ? I tried to solve it like this: x+2(3x-1) but I got the wrong answer and I need to know the method.
Date: 07/23/99 at 08:57:09 From: Doctor Peterson Subject: Re: Algebraic operations Hi, Melanie. When we write a function like f(x) = x+2, that means that whatever value you are given for x is to be substituted for x in the expression. Rather than solve your own problem, I'll work out g(f(x)) instead. It means we have to substitute f(x) for x: g(f(x)) = 3 * f(x) - 1 = 3(x+2) - 1 You could also think of it this way: g(f(x)) = g(x+2) = 3(x+2) - 1 Either way, we can then simplify it to g(f(x)) = 3x + 5 Now let's think about this: if we substitute a particular value for x, such as 2, what will we get? g(f(2)) = 3*2 + 5 = 11 If we just plug x = 2 into the functions one at a time, we should get the same answer: f(2) = 2 + 2 = 4 g(4) = 3*4 - 1 = 11 Composition of functions can be thought of as plumbing: the x first goes into the f machine, and what comes out goes into the g machine: +---+ +---+ 2 --->| f |----> 4 ---->| g |---> 11 +---+ +---+ +---+ +---+ x --->| f |---> x+2 --->| g |---> 3x+5 +---+ +---+ I hope that clarifies what's going on. Now you can connect f to the output of g and find the answer for f(g(x)). - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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