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Unformatted text preview: d the work correctly and we do indeed have the inverse.
Before we move on we should also acknowledge the restrictions of x ³ 0 that we gave in the
problem statement but never apparently did anything with. Note that this restriction is required to
make sure that the inverse, g -1 ( x ) given above is in fact one-to-one.
Without this restriction the inverse would not be one-to-one as is easily seen by a couple of quick
evaluations. g -1 (1) = (1) + 3 = 4 g -1 ( -1) = ( -1) + 3 = 4 2 2 Therefore, the restriction is required in order to make sure the inverse is one-to-one.
The next example can be a little messy so be careful with the work here. Example 3 Given h ( x ) = x+4
find h -1 ( x ) .
2x - 5 Solution
The first couple of steps are pretty much the same as the previous examples so here they are, x+4
2x - 5
y= Now, be careful with the solution step. With this kind of problem it is very easy to make a
mistake here. x ( 2 y - 5) = y + 4 2 xy - 5 x = y + 4
2 xy - y = 4 + 5 x ( 2 x - 1) y = 4 + 5 x
y= 4 + 5x
2x -1 So, if we’ve done all of our work correctly the inver...
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- Spring '12