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Proving
that Two Functions are Sections: Definition / Inverting a graph, Is the inverse a function?, Finding inverses, Proving inverses I have shown how to draw an inverse, given the graph, and how to find an inverse, given the formula. But suppose you are given two functions and told to verify (check) that they are inverses of each other. How would you do that? First, you would need to note that drawing the graphs is not a "proof". To emphasize this, the instructions will often tell you to "verify algebraically" that the functions are inverses. How do you do that? If you think back to the definition of an inverse, the point of the inverse is that it's backwards from what you started with; it takes you back to where you started from. For instance, if the point (1, 3) is on the graph of the function, then the point (3, 1) is on the graph of the inverse. That is, if you start with x = 1, you will go to y = 3; then you plug this into the inverse, and you'll go right back to x = 1, where you started from. It is this property that you use to prove (or disprove) that functions are inverses of each other. You will compose the functions (that is, plug x into one function, plug that function into the inverse function, and then simplify) and verify that you end up with just "x". Here's what it looks like:
Plug g(x) into f (x):
Plug f (x) into g(x) : Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
Both ways, I ended up with just "x", so f (x) and g(x) are inverses of each other.
Plug g(x) into f (x):
I didn't end up with just "x", so f (x) and g(x) are not inverses of each other. This example points out something that can cause
problems for some students. Since the inverse "undoes" whatever the original function
did to x,
the instinct is to create an "inverse" by applying reverse operations. In this case,
since f (x)
multiplied x
by 3
and then subtracted 2
from the result, the instinct is to think that the inverse would be to divide x
by 3
and then to add 2
to the result. But as you saw above, this is not correct. Comparing this example with the previous
example, you can see that the reversed operations were correct, but that they also need to be applied
in reverse order. That is, since Also, in the second example, as soon as I did not end up with "x", I knew the functions were not inverses. I had done the composition ( f o g)(x) and had come up with something other than "x", so I didn't bother checking (g o f )(x). In the first example, however, I checked ( f o g)(x) and came up with "x", and then I checked (g o f )(x), too. Why? Here's an example of why:
Plug g(x) into f (x):
Since I started by plugging x into g(x), then I started with non-negative x-values. Since the absolute value of zero is zero and the absolute value of a positive number is just itself, then, in this case, I can simplify | x | as just "x". Then we have ( f o g)(x) = x. Looking good so far. Now plug f (x) into g(x):
Hmm... Since I started by plugging x into f (x), then I was starting with any value of x. That is, x might have been positive, but it might also have been negative. Since I don't know if x is negative or positive, then I can't remove the absolute-value bars on the final answer, and I'm stuck with an answer of "(g o f )(x) = | x |". So (g o f )(x) does not simplify to x. The answer is: g(x) and f (x) are not inverses of each other. This is why you need to check both ways: sometimes there are fiddly considerations, usually involving square roots, that force the composition not to work, because the domains and ranges of the two functions aren't compatible. In this case, if f (x) had been restricted to non-negative x, then the functions would have been inverses. In general, though, if one composition gives you just "x", then the other one will, too, especially if you're not dealing with restricted domains. But you should remember to do both compositions on tests and such, in order to get full credit. << Previous Top | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Return to Index
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are inverses of each other.
