Haberman MTH c Section I: Sets and Functions Module 6: Inverse Functions EXAMPLE: Consider the unction n ( ), which is graphed in Figure below Figure : Graph o n( ) Since n is a unction, each input corresponds to eactl one output In this case it is also true that each output corresponds to eactl one input For eample, i n ( ) 8 then must be There are no other values that correspond with the output 8 Similarl, i n ( ) 7 then must be Onl the value corresponds with the output 7 We sa that unctions like one-to-one, (sometimes denoted ) n ( ) are DEFINITION: A unction is one-to-one i each output corresponds to eactl one input, ie, i ( a) ( b) then a b
EXAMPLE: Now consider the unction p ( ), which is graphed in Figure below Figure : Graph o p( ) Here, it is not the case that each output corresponds to eactl one input, so p is not oneto-one (I anthing, p is two-to-one ) For eample, i be or p ( ) 4 then ou cannot be sure what is: could Similarl, i p ( ) 6 then could be 4 or 4 We can determine i a unction is one-to-one b appling the horizontal line test to the graph o the unction The Horizontal Line Test I a unction is one-to-one then no horizontal line intersects the graph o the unction more than once EXAMPLE: Which o the ollowing graphs represent one-to-one unctions? Figure Figure 4 Figure 5 SOLUTION: The onl graph that represents a one-to-one unction is the one in Figure The graphs in Figures 4 and 5 do not represent one-to-one unctions because the do not pass the horizontal line test
I a unction ( ) is one-to-one, then, in addition to having the outputs a unction o the inputs, it is also true that the inputs are a unction o the outputs, ie, is a unction o, ie, can pla the role o the independent variable and can pla the role o the dependent variable (This is equivalent to saing that i ou turn our head 90 when looking at the graph o, the graph ou see passes the vertical line test and is a unction Thus, we see that the horizontal line test is just the vertical line test ater reversing the roles o and ) Inverse Functions I a unction is one-to-one then its inverse is a unction The unction that reverses the roles o and is the inverse unction o and is denoted CAUTION: The notation eponent The notation represents the inverse o the unction The is NOT an does NOT mean EXAMPLE: I t and t are inverses o one another, then t () implies that t () Since the inverse o a unction reverses the role o the inputs and the outputs, to ind the inverse o a unction, all that we have to do is reverse the roles o and EXAMPLE: Find the inverse o the unction SOLUTION: n ( ) First we write n in equation orm: solving the equation or : We can then reverse the roles o and b
4 The equation represents the inverse, but we would like to use standard unction notation to represent the inverse Since usuall is used to represent the independent variable, we write n ( ) In Figure 6 below both see, the graph o n and n are graphed, along with the line n is the graph o n relected about the line As ou can Figure 6: n( ) is graphed in blue and n ( ) is graphed in green and the line is graphed in red KEY POINT: The graph o the inverse o a unction is the same as the graph o the original unction but relected about the line EXAMPLE: Find the inverse o the unction ( ) SOLUTION: To ind the inverse o solve or ( ) we irst write the unction as and then
5 ( ) ( ) + + Since we usuall use as our independent variable, we switch our and and obtain the inverse unction ( ) + Recall that the inverse o a unction reverses the roles o and Thus, i ou compose with its inverse ou take to and then to, so is sent to itsel, ie, the composition o a unction with its inverse is the identit unction Inverse Function Test I ( g)( ) and ( g )( ), then the unctions and g are inverses o one another
6 EXAMPLE: Check that ( ) and the unction that we ound in the eample above are in act inverses o one another b perorming the inverse unction test SOLUTION: In the eample above we determined that the inverse o ( ) is ( ) + To check this conclusion we can perorm the inverse unction test To perorm the test, we ( ) ( ) need to veri that ( ) and ( ) ( ) ( ) ( ) ( ) + + + + AND ( ) ( ) ( ( ) ) + + 6 + ( ) Since ( )( ) and ( ) ( ) ( ), + ( ) is in act the inverse o