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Inverse Functions

Date: 08/09/2002 at 18:15:16
From: Andy Provan
Subject: Inverse functions

Hi,

I need help on two algebra problems. I found the answer, but I don't 
know how they got it. Since you have been a big help to me before, I 
was hoping that you could help me out.

Here they are:

   F(x) = 5x-6, G(x) = x-4
   1). (f composed of g)^-1

   F(x) = 5x-6, G(x) = x-4
   2). G^-1 composed of f^-1

Both of these problems have an answer of x+26/5. Would you please 
explain at least one of them for me?

Thanks again,
Andy


Date: 08/09/2002 at 23:11:41
From: Doctor Peterson
Subject: Re: Inverse functions

Hi, Andy.

Your terminology needs a little correction; "f o g" is "f composed 
WITH g" not "OF g" , which would mean that f is made out of g's. 
Rather, f and g are "put together," which is what "composed" means. 
But I've seen enough variation in the use of math words around the 
world that I may well be wrong.

Let's first compose f with g in the first problem:

    fog(x) = f(g(x)) = f(x-4) = 5(x-4)-6 = 5x - 26

That means we replace "x" in the definition of f with the value of 
g(x), and then simplify.

Now we have to invert this. If y is the value of fog(x), we have

    y = 5x - 26

Solving for x,

    y + 26 = 5x

    (y + 26)/5 = x

So the inverse of fog, which takes y back to x, is

    (fog)^-1(y) = (y + 26)/5

Since the variable in a function definition is just a placeholder, we 
can replace y with x and get the answer you have (with parentheses 
where they belong).

The second problem is similar, but you will invert each function 
first, then compose.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Trigonometry

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