Mth141S51 - Sec 5.1 Inverse Functions The word inverse also...

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Sec. 5.1 – Inverse Functions The word “ inverse ” also has a similar meaning to the word “ reverse ”. Reverse can mean to do the opposite. To go somewhere in your car you follow a certain path. To return back home you can just follow the path backwards or in the opposite direction. Just do the steps backwards. Where you had turned right to get to your destination, you will turn left at that location when you come back home. And so on. In mathematics, we have inverse elements for some operations. For example, in addition we know that 3 + (-3) = 0 . By adding the inverse ( opposite) of 3, we cancel the 3 and get 0. You use this technique many times in solving equations . For example in 3 + 7x = 17, we would like to eliminate the 3 on the left. We can do that by adding a -3 to both sides of the equation. Since -3 + 3 equals 0, the additive identity element, the left side becomes 0 + 7x or, of course, just 7x. So we eliminated the 3 by adding its opposite (or its inverse). Now in the resulting equation 7x = 14, we would like to eliminate the 7 so we just have x. We can do this if we multiply by its inverse element (reciprocal) on both sides of the equation. . Since 7 1 1 1 7 = , the multiplicative identity element, the left side becomes 1x, or just x. And the resulting equation is x = 2. Now an important principle here is that if you combine a number and its inverse value, you will get the identity element for that operation!! A number plus its additive inverse (opposite) gives you 0 and a number times its multiplicative inverse (reciprocal) gives you 1. We will revisit this idea later. Now in section 4.1, the authors start out by discussing inverse relations ! Note not functions! In a relation, we find the inverse values by interchanging the x and y values. If you have a set of ordered pairs, say {(3,2) , (4,7) , (1,5)}, then the inverse of this relation would be {(2,3) , (7,4) , (5,1)}. You simply interchange the first and second members. If the relation is indicated by an equation, we find the inverse equation by simply interchanging the x and y variables. So if the relation is defined as 2 2 3 y y x + - = , then the inverse relation would be 2 2 3 x x y + - = . If the relation is indicated by our Venn diagram mapping of domain values on the left to range values on the right, then the inverse would be mapping the values on the right to the values on the left. Again, we are simply “reversing” the values. Functions and inverses Now we would like to apply the idea of an inverse to functions !
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In order to do this we encounter a slight problem, two values in the domain could pair up with the same value in the range. This is ok for a function. However, if we are to reverse the connection, then the one value in the second group now will pair up with two values in the first group. This violates the definition of a function. So the “inverse function” would NOT be a function at all! So for the inverse of a function to be a function, we cannot have more than one value in the original domain pairing up with a value in the
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This note was uploaded on 06/06/2011 for the course MTH mth141 taught by Professor Stean during the Spring '10 term at Moraine Valley Community College.

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Mth141S51 - Sec 5.1 Inverse Functions The word inverse also...

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