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Question

g(x)=(x-3)/11

h={(-3,-8),(3,9),(4,-3),(9,0)}

 

a. g^-1(x)=

b.(gog^-1)(0)=

h^-1(9)=

Answer 1

A)  To find the inverse of g(x), these steps work well.

 

1.  Write down the problem.

 

      g(x) = (x - 3)/11

 

2.  Replace the function notation, in this case g(x), with the letter variable y.  (It's easier to work with.)

 

       y = (x - 3)/11

 

3.  Switch x and y.  (That is, rewrite the equation, but for every y, write x; and for every x, write y.)

 

      x = (y - 3)/11

 

4.  Solve for y.  (That is, get y by itself on one side of the equation such that the other side is just a bunch of x's and numbers.....no y's.)

 

      x = (y - 3)/11                                              Multiply both sides by 11 to get:

 

      11x = y - 3                                                 Add 3 to both sides to get y by itself.

 

      11x + 3 = y                                                Which is the same as

 

      y = 11x + 3

 

5.  Replace y with the function notation, in this case g^-1(x).

 

       g^-1(x) = 11x + 3

 

Done!

 

 

B.  Two ways to do this:

 

(gog^-1)(0) is telling you to first plug 0 into  g^-1(x), then take the result and plug that into g(x).

 

So, plug 0 into  g^-1(x), i.e,  g^-1(0) = 11(0) + 3.

 

That simplifies to  g^-1(0) = 3.

 

Then plug that result into g(x), i.e. g(3) = (3 - 3)/11 

 

That simplifies to g(3) = 0.

 

So (gog^-1(0)) = 0

 

The second way to do this is to remember that gog^-1(x) = x for all functions with defined inverses.  Basically that means the g and the g^-1 cancel each other out, leaving you with only x left.  So gog^-1(0) = 0.

 

C)  When you have a function defined as a set of points, that basically means that the inverse function just switches all the x's and y's with each other.  

 

As a result, h(x) = {(-3,-8),(3,9),(4,-3),(9,0)} becomes h^-1(x) = {(-8,-3),(9,3),(-3,4),(0,9).

 

To find h^-1(9), use h^-1(x) and find the point with x-coordinate 9, i.e. (9,3).  h^-1(9) is just the y-coordinate of that point, i.e. 3.

 

Therefore h^-1(9) = 3