One-to-one

1 Inverse Functions Inverse Functions One-to-one One-to-one Suppose fff ::: A. Suppose A . A !B. * B isis aaa function. B function. We function. We call fff One-to-one We call call One-to-one ifif every every distinct distinct pair of pair of objects objects in A isis assigned in A. A assigned to to aaa distinct to distinct pair distinct pair of pair of objects objects in B. In in B. In other other words, each words, each object object of of the target has the target target has at has at most at most one most one object one object from from the the domain domain assigned assigned to to it. it. There There isis aa way way of of phrasing phrasing the previous definition the previous previous definition in definition in aa more more mathematical mathematical language: fff isis One-to-one language: language: One-to-one ifif whenever whenever we we have we have two have two objects 2e AA with objects a,a,cc with aa =. ~ c, 6= c, we are c, we are guaranteed guaranteed thatthat fff(a).

2 (a) =. (a) $ fff(c). 6= (c). (c). Example. fff ::: R. Example. Example. R !R. IR R. JR where * where fff(x). (x). (x) =. = x = xx222 is not One-to-one is not not One-to-one because because 33 =. 6 ~ 3. 3. and and yet fff(3). yet and yet (3). (3) = ff( 3). = f = (( 3) since fff(3). 3) since (3) and fff( 3). (3) and and (( 3) 3) both equal 9. both equal 9. Horizontal line Horizontal line test test If aa horizontal If horizontal line line intersects intersects the the graph graph of of fff(.x) in more (x) in (x) more than than one one point, point, then ff(z). then f(x) is (x) is not is not not One-to-one . One-to-one . reason ff(x). reason The reason The f(x) would not (x) would not bebe One-to-one One-to-one is that the is that the graph graph would would contain contain two points two that points that have the that have the same same second second coordinate coordinate for for example, example, (2, (2,3) and 3) and (4,3). (4,3). (4, That 3). That would That would would mean that fff(2).

3 Mean that and fff(4). (2) and (2) (4) both (4) equal 3, both equal 3, and and One-to-one One-to-one functions can't functions can't assign can't assign two di erent two different objects in different objects in the the domain domain to to the the same same object object of the of the target. target. every horizontal If every If horizontal line line inin RJR222 intersects R intersects the the graph graph ofof aa function function at most at most once, once, then once, then the the function function is is One-to-one . One-to-one . R. of ff ::: R !R R where ff(z) 22. Examples. Below Examples. Below is is the the graph graph of JR R where , (x) = z2. = x . There There isis aa horizontal line horizontal line that that intersects this graph intersects this graph in in more more than than one one point, point, soso ff isis not not One-to-one . One-to-one . \~. )L2. 90. 66. 66. Below is the graph of g R 1W where gQr) = x3. Ally horizontal line that ~. 3. Below Below could be is Below Below is drawn is the isthe the the graph would graph graph of graph ofg : gg ::gR.

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7 The The function function function function most ggg ghas one point, has has has the the the the so g is set set set set R. R for for R1 Wfor forits One-to-one its its range. range. itsrange. This This equals equals Thisequals the the equalsthe target target thetarget of of g, g, so so g g is is targetofofg,g,sosog gisisonto. onto. onto. onto. * * * * * * * * * * * * *. Onto *. ** ** ** ** ** ** 67** ** ** ** * ** * ** * ** *. Suppose f : A B is ~ a*function. 67 We call f onto if the range of 91. 67. 67. f equals 67. B. In other words, f is onto if every object in the target has at least one object from the domain assigned to it by f. What an inverse function is Suppose f : A ! B is a function. A function g : B ! A is called the inverse function of f if f g = id and g f = id. If g is the inverse function of f , then we often rename g as f 1 . Examples. Let f : R ! R be the function defined by f (x) = x + 3, and let g : R ! R be the function defined by g(x) = x 3.

8 Then f g(x) = f (g(x)) = f (x 3) = (x 3) + 3 = x Because f g(x) = x and id(x) = x, these are the same function. In symbols, f g = id. Similarly g f (x) = g(f (x)) = g(x + 3) = (x + 3) 3=x so g f = id. Therefore, g is the inverse function of f , so we can rename g 1. as f , which means that f 1 (x) = x 3. Let f : R ! R be the function defined by f (x) = 2x + 2, and let g : R ! R be the function defined by g(x) = 12 x 1. Then 1 1 . f g(x) = f (g(x)) = f x 1 =2 x 1 +2=x 2 2. Similarly 1 . g f (x) = g(f (x)) = g(2x + 2) = 2x + 2 1=x 2. Therefore, g is the inverse function of f , which means that f 1. (x) = 12 x 1. 92. The Inverse of an inverse is the original If f 1 is the inverse of f , then f 1 f = id and f f 1 = id. We can see from the definition of inverse functions above, that f is the inverse of f 1 . That is (f 1 ) 1 = f . Inverse functions reverse the assignment . The definition of an inverse function is given above, but the essence of an inverse function is that it reverses the assignment dictated by the original function.

9 If f assigns a to b, then f 1 will assign b to a. Here's why: If f (a) = b, then we can apply f 1 to both sides of the equation to obtain the new equation f 1 (f (a)) = f 1 (b). The left side of the previous equation involves function composition, f 1 (f (a)) = f 1 f (a), and f 1 f = id, so we are left with f 1 (b) = id(a) = a. The above paragraph can be summarized as If f (a) = b, then f 1 (b) = a.. Examples. 1. If f (3) = 4, then 3 = f (4). 1. If f ( 2) = 16, then 2=f (16). 1. If f (x + 7) = 1, then x + 7 = f ( 1). 1. If f (0) = 4, then 0 = f ( 4). 1. If f (x2 3x + 5) = 3, then x2 3x + 5 = f (3). In the 5 examples above, we erased a function from the left side of the equation by applying its inverse function to the right side of the equation. When a function has an inverse A function has an inverse exactly when it is both One-to-one and onto. This will be explained in more detail during lecture. * * * * * * * * * * * * *. 93. Using inverse functions Inverse functions are useful in that they allow you to undo a function.

10 Below are some rather abstract (though important) examples. As the semes- ter continues, we'll see some more concrete examples. Examples. Suppose there is an object in the domain of a function f , and that this object is named a. Suppose that you know f (a) = 15. If f has an inverse function, f 1 , and you happen to know that f 1 (15) = 3, then you can solve for a as follows: f (a) = 15 implies that a = f 1 (15). Thus, a = 3. If b is an object of the domain of g, g has an inverse, g(b) = 6, and 1. g (6) = 2, then b = g 1 (6) = 2. 1. Suppose f (x + 3) = 2. If f has an inverse, and f (2) = 7, then 1. x+3=f (2) = 7. so x=7 3=4. * * * * * * * * * * * * *. The Graph of an inverse If f is an invertible function (that means if f has an inverse function), and if you know what the graph of f looks like, then you can draw the graph of f 1. If (a, b) is a point in the graph of f (x), then f (a) = b. Hence, f 1 (b) = a. That means f 1 assigns b to a, so (b, a) is a point in the graph of f 1 (x).