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Finding an Inverse Algebraically  Problem 2
Carl Horowitz
Carl Horowitz
University of Michigan
Runs his own tutoring company
Carl taught upperlevel math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
Finding the inverse of a function algebraically. Now whenever we are finding the inverse of a function the first thing we always have to do is make sure our function is 1 to 1. So looking at the function we have right now x² plus 2 we know what the graph of x² plus x² is, and we know the plus 2 just shifted up 2.
So for this particular graph we know that we are up 2 and a basic parabola, something like this. First thing I know is that parabola is not 1 to 1, for every y is there only one x? No there’s not so as is this function does not have the inverse. What you can do though is restrict your domain and what I mean by that is, is there a way to make it so this function is actually 1 to 1. If we look at it, if we just chop off either one of these sides, so if I say we are only looking at positive x’s, then the graph is 1 to 1 for every y there’s only one x because this side doesn’t exist.
You could do the other side as well typically people like dealing with positive numbers more than negative numbers, so what I’m actually going to do is just restrict the domain for only positive x’s. If I chop it off here, I erase this position I now have a 1 to 1 function. So what I can say is x has to be greater and equal to 0, what I’ve done is I’ve taken my function which wasn’t 1 to 1 and turned it into a 1 to 1 function. It’s called the domain restriction.
From here how we find the inverse is how we would in the other inverse. So g of x is the same thing as y, this becomes y is equal to x² plus 2, switch our x and our y’s and then solve for y, subtract 2 and take the square root. Throw in our inverse notation so then y becomes f inverse of x and then well after that square root of x minus 2.
What we also need to be careful about though is our domain restriction, remember that we restricted our x’s to make this 1 to 1. We still need to restrict what goes into this function. This is our domain over here, so it turns into a range down here. So we need to think about what values of x we can put in here. The values of x that go into a inverse are the values of the range that started, remember the values of x in here correspond to the values of y up here. Switch our x’s and y’s, switch our domains and range.
So what we are concerned with then is range of our initial graph. Range corresponds to the y value, we have a parabola we moved it up 2 so our range is actually 2 and up for our original graph. Take our original range that becomes our domain for inverse. So what we need throw in here is x has to be greater than equal to 2 to correspond to the original graph or the original graph’s range for a new domain. We can also just sort of think about this one a little bit more logically looking at our graph square root of x minus 2, we need to plug in values such that we can take the square root of it, we can’t take the square root of negative numbers so we are left with has to be greater or equal to 2.
So finding the inverse of a function that is not 1 to 1 requires a little bit more work just making sure you restrict your domain and then finding the right domain for your inverse function as well.
So finding the inverse of a function that is not 1 to 1 requires a little bit more work just making sure you restrict your domain and then finding the right domain for your inverse function as well.
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Carl Horowitz
B.S. in Mathematics University of Michigan
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