SOLUTION: The one-to-one functions g and h are defined as follows. g={(-7,-6), (-5,4), (4,-7)(7,6)} h(x)= 3x-14 Find the following. g^-1(4)= h^-1(x)= (h^-1 o h)(7)=

The one-to-one functions g and h are defined as follows. 
 g={(-7,-6), (-5,4), (4,-7)(7,6)}
 h(x)= 3x-14 
 Find the following. 
 g-1(4)=

"g-1(4)" just says "Find the pair of coordinates that has 4 for its
 y-coordinate, and the answer is its x-coordinate".  So we look through those
 and find (-5,4) is the only one of those up there that has a 4 for it's y-
coordinate, and so its x-coordinate is -5 and we write:

g-1(4)=-5   

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h-1(x)= 

Start with
    h(x) = 3x-14
Change "h(x): to "y"
       y = 3x-14
Interchange x and y:
       x = 3y-14
Solve for y:
    x+14 = 3y
(x+14)/3 = y  
Change y to h^-1(x)

h-1(x) = (x+14)/3 
-------------------------
 (h-1 o h)(7)=

That's the same as:

h-1(h(7)) = ?

First find h(7)
   h(x) = 3x-14
   h(7) = 3(7)-14
   h(7) = 21-14
   h(7) = 7

Then h-1(h(7)) = h-1(7) = 

then find h-1 of 7
by plugging 7 in for x.

h-1(x) = (x+14)/3
h-1(7) = (7+14)/3
h-1(7) = 21/3
h-1(7) = 7

h-1(h(7)) = h-1(7) = 7.  So when a function is
composed with its inverse, you get a function that gives you back the same
number for y that you substituted for x.  That's analogous to 7x1=7, like
multiplying by 1 or adding 0.  

Edwin