It is very important not to confuse function notation with negative exponents.

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1 Inverse Functions What is an Inverse Function? An inverse unction is a unction that will undo anything that the original unction does. For example, we all have a way o tying our shoes, and how we tie our shoes could be called a unction. So, what would be the inverse unction o tying our shoes? The inverse unction would be untying our shoes, because untying our shoes will undo the original unction o tying our shoes. Let s look at an inverse unction rom a mathematical point o view. Consider the unction (x) = 2x 5. I we take any value o x and plug it into (x) what would happen to that value o x? First, the value o x would get multiplied by 2 and then we would subtract 5. The two mathematical operations that are taking place in the unction (x) are multiplication and subtraction. Now let s consider the inverse unction. What two mathematical operations would be needed to undo (x)? Division and addition would be needed to undo the multiplication and subtraction. A little arther down the page we will ind the inverse o (x) = 2x 5, and hopeully the inverse unction will contain both division and addition (see example 5). Notation I (x) represents a unction, then the notation inverse o (x). Similarly, the notation g(x). g (x), (x), read inverse o x, will be used to denote the read g inverse o x, will be used to denote the inverse o Note: (x). (x) 1 It is very important not to conuse unction notation with negative exponents. Does the Function have an Inverse? Not all unctions have an inverse, so it is important to determine whether or not a unction has an inverse beore we try and ind the I a unction does not have an inverse, then we need to realize the unction does not have an inverse so we do not waste time trying to ind something that does not exist. So how do we know i a unction has an inverse? To determine i a unction has an inverse unction, we need to talk about a special type o unction called a One to One Function. A one to one unction is a unction where each input (x value) has a unique output (y value). To put it another way, every time we plug in a value o x we will get a unique value o y, the same y value will never appear more than once. A one to one unction is special because only one to one unctions have an inverse unction. Note: Only One to One Functions have an inverse unction. Examples Now let s look at a ew examples to help demonstrate what a one to one unction is. Example 1: Determine i the unction = {(7, ), (8, 5), ( 2, 11), ( 6, 4)} is a one to one unction. The unction is a one to one unction because each o the y values in the ordered pairs is unique; none o the y values appear more than once. Since the unction is a one to one unction, the unction must have an

2 Example 2: Determine i the unction h = {(, 8), ( 11, 9), (5, 4), (6, 9)} is a one to one unction. The unction h is not a one to one unction because the y value o 9 is not unique; the y value o 9 appears more than once. Since the unction h is not a one to one unction, the unction h does not have an Remember that only one to one unction have an Is the Function a One to One Function? We can determine i unctions are one to one by looking at ordered pairs and determining i each o the y values is unique, but what i we do not have ordered pairs? We could create ordered pairs by plugging dierent x values into the unction and inding the corresponding y values giving us some ordered pairs. Rather than spending time creating ordered pairs, why not consider looking at the entire graph o the unction instead? By looking at the entire graph rather than a ew points, we should still be able to determine i the unction is a one to one unction or not. In looking at the graph o the unction we can determine i a unction is a one to one unction or not by applying the Horizontal Line Test, or HLT. I the graph o the unction passes the Horizontal Line Test, then the unction is a one to one unction. I the graph o the unction ails the Horizontal Line Test, then the unction is not a one to one unction. Horizontal Line Test The HLT says that a unction is a one to one unction i there is no horizontal line that intersects the graph o the unction at more than one point. By applying the Horizontal Line Test not only can we determine i a unction is a one to one unction, but more importantly we can determine i a unction has an inverse or not. Examples Now let s look at a ew examples to help explain the Horizontal Line Test. Example : Determine i the unction (x) = x + 2 is a one to one unction. 4 To determine i (x) is a one to one unction, we need to look at the graph o (x). Since (x) is a linear equation the graph o (x) is a line with a slope o /4 and a y intercept o (0, 2). In looking at the graph, you can see that any horizontal line (shown in red) drawn on the graph will intersect the graph o (x) only once. (x) is a one to one unction and (x) must have an

3 Example 4: Determine i the unction g(x) = x 4x is a one to one unction. To determine i g(x) is a one to one unction, we need to look at the graph o g(x). In looking at the graph, you can see that the horizontal line (shown in red) drawn on the graph intersects the graph o g(x) more than once. g(x) is not a one to one unction and g(x) does not have an How to Find the Inverse Function Now that we have discussed what an inverse unction is, the notation used to represent inverse unctions, one to one unctions, and the Horizontal Line Test, we are ready to try and ind an inverse unction. Here are the steps required to ind the inverse unction: Step 1: Determine i the unction has an Is the unction a one to one unction? I the unction is a one to one unction, go to step 2. I the unction is not a one to one unction, then say that the unction does not have an inverse and stop. Step 2: Change (x) to y. Step : Switch x and y. Step 4: Solve or y. Step 5: Change y back to 1 ( x ). By ollowing these 5 steps we can ind the inverse unction. Make sure that you ollow all 5 steps. Many people will skip step 1 and just assume that the unction has an inverse; however, not every unction has an inverse, because not every unction is a one to one unction. Only unctions that pass the Horizontal Line Test are one to one unctions and only one to one unctions have an

4 Examples Now let s use the steps shown above to work through some examples o inding inverse unctions. Example 5: I (x) = 2x 5, ind the This unction passes the Horizontal Line Test which means it is a one to one unction that has an y = 2x 5 Change (x) to y. x = 2y 5 Switch x and y. x + 5 = 2y x + 5 = y 2 x (x) = = x Solve or y by adding 5 to each side and then dividing each side by 2. Change y back to 1 (x). 1 x 5 (x) = + or (x) = x Example 6: I (x) = x 2 + 4, ind (x). This unction does not pass the Horizontal Line Test which means it is not a one to one unction. (x) does not exist (x) is not a one to one, so (x) = x does not have an (x) does not exist.

5 Example 7: I (x) = (x 2), ind the This unction passes the Horizontal Line Test which means it is a one to one unction that has an y = (x 2) Change (x) to y. x = (y 2) Switch x and y. x = (y 2) x = y 2 x + 2 = y Solve or y by taking the cube root o each side and then adding 2 to each side. Change y back to 1 (x). (x) = x + 2. Addition Examples I you would like to see more examples o inding inverse unctions, just click on the link below. Additional Examples Practice Problems Now it is your turn to try a ew practice problems on your own. Work on each o the problems below and then click on the link at the end to check your answers. 4x Problem 1: I (x) =, ind (x). 2x Problem 2: I (x) = x, ind (x). 6 4 Problem : I (x) = (x + 2) 2 1, ind Problem 4: I (x) = x + 11, ind (x). (x). Problem 5: I (x) = 5 x + 2, ind (x). 2x 5 Problem 6: I (x) =, ind (x). Solutions to Practice Problems