Lecture 10_Inverse Function PDF

One-to-One Functions and Their Inverses The inverse of a function is a rule that acts on the output of the function and produces the corresponding input. So the inverse “undoes” or reverses what the function has done. Not all functions have inverses; those that do are called one-to-one....

One-to-One Functions and Their Inverses The inverse of a function is a rule that acts on the output of the function and produces the corresponding input. So the inverse “undoes” or reverses what the function has done. Not all functions have inverses; those that do are called one-to-one. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. One-to-One Functions Let’s compare the functions f and g whose arrow diagrams are shown in Figure 1. f is one-to-one g is not one-to-one Figure 1 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. One-to-One Functions Note that f never takes on the same value twice (any two numbers in A have different images), whereas g does take on the same value twice (both 2 and 3 have the same image, 4). In symbols, g (2) = g (3) but f (x1) ≠ f (x2) whenever x1 ≠ x2. Functions that have this latter property are called one-to-one. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. One-to-One Functions DEFINITION OF A ONE-TO-ONE FUNCTION A function with domain A is called a one-to-one function if no two elements of A have the same image, that is, An equivalent way of writing the condition for a one-to-one function is this: If f (x1) = f (x2), then x1 = x2. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  One-to-One Functions If a horizontal line intersects the graph of f at more than one point, then we see from Figure 2 that there are numbers x1 ≠ x2 such that f (x1) = f (x2). This function is not one-to-one because f (x1) = f (x2). Figure 2 This means that fStewart/Redlin/Watson, is not one-to-one. Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. One-to-One Functions Therefore, we have the following geometric method for determining whether a function is one-to-one. HORIZONTAL LINE TEST A function is one-to-one if and only if no horizontal line intersects its graph more than once. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1 – Deciding Whether a Function Is One-to-One Is the function f ( x ) = x 3 one-to-one? Solution 1: If x1  x2 , then x13  x23 (two different numbers cannot have the same cube). Therefore, f ( x ) = x 3 is one-to-one. Solution 2: From Figure 3 we see that no horizontal line intersects the graph of f ( x ) = x 3 more than once. Therefore by the f ( x ) = x 3 is one-to-one Horizontal Line Test, f is one-to-one. Stewart/Redlin/Watson, Figure Algebra and Trigonometry, 4th Edition. © 2016 Cengage. 3 Reserved. All Rights May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. One-to-One Functions Notice that the function f of Example 1 is increasing and is also one- to-one. In fact, it can be proved that every increasing function and every decreasing function is one-to-one. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Inverse of a Function One-to-one functions are important because they are precisely the functions that possess inverse functions according to the following definition. DEFINITION OF THE INVERSE OF A FUNCTION Let f be a one-to-one function with domain A and range B. Then its inverse function f −1 has domain B and range A and is defined by f −1 ( y ) = x  f ( x ) = y for any y in B. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Inverse of a Function This definition says that if f takes x to y, then f −1 takes y back to x. (If f were not one-to-one, then f −1 would not be defined uniquely.) The arrow diagram in Figure 6 indicates that f −1 reverses the effect of f. From the definition we have Figure Stewart/Redlin/Watson, Algebra and 6 4th Edition. © 2016 Cengage. All Rights Reserved. Trigonometry, May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1 – Finding f inverse for Specific Values If f (1) = 5, f (3) = 7, and f (8) = −10, find Solution: From the definition of f −1 we have f −1 ( 5 ) = 1 because f (1) = 5 f −1 ( 7 ) = 3 because f ( 3 ) = 7 f −1 ( −10 ) = 8 because f ( 8 ) = −10 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1 – Solution Figure 7 shows how f −1 reverses the effect of f in this case. Figure 7 Don’t mistake the −1 in f −1 for an exponent. 1 f ( x ) does not mean −1 f (x) is written as ( f ( x ) ). 1 −1 The reciprocal f (Stewart/Redlin/Watson, x) Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Inverse of a Function By definition the inverse function f −1 undoes what f does: If we start with x, apply f, and then apply f −1, we arrive back at x, where we started. Similarly, f undoes what f −1 does. In general, any function that reverses the effect of f in this way must be the inverse of f. These observations are expressed precisely as follows. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Inverse of a Function INVERSE FUNCTION PROPERTY Let f be a one-to-one function with domain A and range B. The inverse function f −1 satisfies the following cancellation properties: f −1 ( f ( x ) ) = x for every x in A ( ) f f −1 ( x ) = x for every x in B Conversely, any function f −1 satisfying these equations is the inverse of f. These properties indicate that f is the inverse function of f −1, so we say that f and f −1 are inverses of each other. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Finding the Inverse of a Function Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  Finding the Inverse of a Function Now let’s examine how we compute inverse functions. We first observe from the definition of f −1 that y = f ( x )  f −1 ( y ) = x So if y = f (x) and if we are able to solve this equation for x in terms of y, then we must have x = f −1 ( y ). If we then interchange x and y, we have y = f ( x ) , −1 which is the desired equation. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Finding the Inverse of a Function HOW TO FIND THE INVERSE OF A ONE-TO-ONE FUNCTION 1. Write y = f(x). 2. Solve this equation for x in terms of y (if possible). 3. Interchange x and y. The resulting equation is y = f −1 ( x ). Note that Steps 2 and 3 can be reversed. In other words, we can interchange x and y first and then solve for y in terms of x. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 2 – Finding the Inverse of a Function Find the inverse of the function f (x) = 3x − 2. Solution: First, we write y = f (x). y = 3x − 2 Then we solve this equation for x. 3x = y + 2 Add 2 y +2 x= Divide by 3 3 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 2 – Solution Finally, we interchange x and y: x+2 y= 3 x+2 Therefore, the inverse function is f ( x ) = −1. 3 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Finding the Inverse of a Function A rational function is a function defined by a rational expression. In the next example we find the inverse of a rational function. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 3 – Finding the Inverse of a Rational Function 2x + 3 Find the inverse of the function f ( x ) =. x −1 Solution: ( 2x + 3 ) We first write y = and solve for x. ( x − 1) 2x + 3 y= Equation defining function x −1 y (x − 1) = 2x + 3 Multiply by x − 1 yx − y = 2x + 3 Expand Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 3 – Solution yx − 2x = y + 3 Bring x-terms to LHS x (y − 2) = y + 3 Factor x y +3 x= Divide by y − 2 y −2 x +3 f −1 ( x ) =. Interchange x and y x −2 x +3 Therefore, the inverse function is f (x) = −1. x −2 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 4 – Finding the Inverse of a Rational Function Solution: Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. CHECK YOUR ANSWER We use the Inverse Function Property: Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5 – Finding the Inverse of a Rational Function Use the Inverse Function Property to show that f and g are inverses of each other. Solution: Thus, f and g are inverses of each other. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 6 – Finding the Inverse of a Rational Function Use the Inverse Function Property to show that f and g are inverses of each other. Solution: Thus f and g are inverses ofAlgebra Stewart/Redlin/Watson, eachandother. Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Graphing the Inverse of a Function Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  Graphing the Inverse of a Function The principle of interchanging x and y to find the inverse function also gives us a method for obtaining the graph of f −1 from the graph of f. If f (a) = b, then f −1 ( b ) = a. Thus, the point (a, b) is on the graph of f if and only if the point (b, a) is on the graph of f −1. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Graphing the Inverse of a Function But we get the point (b, a) from the point (a, b) by reflecting in the line y = x (see Figure 9). Therefore, as Figure 10 illustrates, the following is true. The graph of f −1 is obtained by reflecting the graph of f in the line y = x. Figure 9 Figure 10 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 7 – Graphing the Inverse of a Function (a) Sketch the graph of f ( x ) = x − 2. −1 (b) Use the graph of f to sketch the graph of f. −1 (c) Find an equation for f. Solution: (a) We sketch the graph of y = x − 2 by plotting the graph of the function y = x and shifting it to the right 2 units. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 7 – Solution (b) The graph of f −1 is obtained from the graph of f in part (a) by reflecting it in the line y = x, as shown in Figure 11. Figure 11 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 7 – Solution (c) Solve y = x − 2 for x, noting that y  0. x −2 = y x − 2 = y2 Square each side x = y2 + 2 y  0 Add 2 Interchange x and y, as follows: y = x2 + 2 x  0 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  Example 7 – Solution Thus f −1 ( x ) = x 2 + 2 x  0 This expression shows that the graph of f −1 is the right half of the parabola y = x 2 + 2, and from the graph shown in Figure 11 this seems reasonable. Figure 11 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.