Function Inverses In an earlier lesson, we introduced functions by assigning US citizens a social security number (SSN) or by assigning students and staff at Sam Houston State University a student ID (SamID) Elementary Functions The function SamID assigns to each student or staff a nine digit number that begins with three zeroes. The Sam ID of the authors of these notes is Part 1, Functions 000354765, so SamID(Ken W. Smith) = 000354765: Lecture 1.6a, Function Inverses In practice, it is important not only that SamID be a function, but that the function process can be reversed. Computer databases at Sam Dr. Ken W. Smith Houston allow a staff member to type in the Sam ID and pull up information about the student/staff member. Each Sam ID number is Sam Houston State University linked to a unique student/staff member! 2013 This is the concept of an inverse function. If SamID maps student/staff to numbers, the inverse function, SamID−1 maps numbers to student/staff. (So SamID−1(000354765) = Ken W. Smith.) Smith (SHSU) Elementary Functions 2013 1 / 33 Smith (SHSU) Elementary Functions 2013 2 / 33 Function Inverses Function Inverses The meaning of function inverse In many applications, we need to reverse the function process, asking for the input x associated with an output y = f(x): Given a function f(x), with inputs x and outputs y, we would like to reverse the process, taking outputs y and restoring the original input x to create f −1(x): In the picture above, we have a function f from the set X (the domain) to Y (the codomain.) We would like to create an inverse function with domain Y that maps back to X. We write f −1 for the new function that reverses the process of function f. 1 Note that f −1 is NOT the reciprocal of f(x). It is NOT ! f Smith (SHSU) Elementary Functions 2013 3 / 33 Smith (SHSU) Elementary Functions 2013 4 / 33 Function Inverses Function Inverses Suppose f : fa; b; cg ! f1; 2; 3g is described in the first picture below (from Wikipedia.) Then f −1 is described by the second picture. Notice how f −1 reverses the inputs and outputs. Warning! The superscript −1 indicates the inverse function. 1 f −1 is not the same as : f Not every function has an inverse. We will look at some examples of functions where we can reverse the process and some examples where we cannot. Smith (SHSU) Elementary Functions 2013 5 / 33 Smith (SHSU) Elementary Functions 2013 6 / 33 Function inverses, examples Function inverses, examples Examples. 1 Let f(x) = 3x + 5: Write this function in the form y = 3x + 5: 2 Suppose we are given a particular output y. Can we recover x? 2 f(x) = x with domain R and codomain R. Yes. Let's take the equation y = 3x + 5 and solve for x: If we write y = x2, and we are given a particular value of y, say y − 5 y = 3x + 5 =) y − 5 = 3x =) = x: y = 25, can we reverse the process and find x? 3 No, not in this case, for both x = −5 and x = 5 are mapped to y−5 We have discovered that if we are given y then x = 3 : y = 25 by this function. y − 5 There are two different inputs that are both mapped to 25 so we g(y) = : 3 cannot reverse our process in a unique way. We follow custom and use x for inputs and y for outputs so we write The function f(x) = x2, with domain , does not have an inverse x − 5 R g(x) = or function. 3 x − 5 f −1(x) = : 3 Smith (SHSU) Elementary Functions 2013 7 / 33 Smith (SHSU) Elementary Functions 2013 8 / 33 Changing the domain to create an inverse function Occasionally, if a function does not have an inverse, we may be able to change the domain of the function so that on this new domain the function is invertible. We can do that in this last example. Elementary Functions Part 1, Functions We could change our function so that the domain is the interval [0; 1) Lecture 1.6b, Computing Function Inverses instead of (−∞; 1): If we agree that no negative numbers are input into this function, then the ambiguity about x goes away. Dr. Ken W. Smith If y = 25 then x must be equal to 5, not −5: In this case, if f : [0; 1) ! (−∞; 1) is defined by f(x) = x2 then the Sam Houston State University p inverse function is f −1(x) = x: 2013 In the next presentation we will practice inverting functions. (END) Smith (SHSU) Elementary Functions 2013 9 / 33 Smith (SHSU) Elementary Functions 2013 10 / 33 Computing function inverses Finding an inverse function by reversing the operations A function f : D ! C has an inverse f −1 : C ! D if and only if f is both a one-to-one function and an onto function. Sometimes it is obvious that function has an inverse. Sometimes a function may be defined in terms of a sequence of operations and each of If the function f is onto then every element of C has at least one them is reversible. preimage back in D. For example, consider the function f(x) = x + 4: If the function f is one-to-one then every element of C which has a What does f do to an input? preimage has a unique preimage. (Recall that this uniqueness is the critical Easy { it adds 4 to every input. part of the definition of a function!) How would one reverse that? −1 (A separate presentation will focus more on one-to-one and onto By subtracting 4 from every input. So f (x) = x − 4: functions.) Smith (SHSU) Elementary Functions 2013 11 / 33 Smith (SHSU) Elementary Functions 2013 12 / 33 Finding an inverse function by reversing the operations Finding an inverse function by reversing the operations One more example, the more complicated invertible function seen earlier: suppose f(x) = 3x + 5: Consider the function f(x) = 3x: What is the inverse function? What does f do to every input? The function f(x) = 3x + 5 first multiplies the input by 3 and then adds It multiplies each input x by 3: 5. How would one reverse \multiplication by 3"? So to reverse this function process we should first subtract 5 and then By dividing by 3. divide by 3. So the inverse function must divide each input by 3. x − 5 So the inverse function should be f −1(x) = : −1 x 3 With this simple logic, we see that f (x) = 3 : (Notice that when we reverse the processes here, we also reversed the order of the operations. The function f multiplies and then adds so f −1 must first subtract and then divide.) Smith (SHSU) Elementary Functions 2013 13 / 33 Smith (SHSU) Elementary Functions 2013 14 / 33 Finding an inverse function algebraically Function inverses, worked examples Some worked exercises. Use algebra to find the inverse function of each function given below. If the function f(x) is defined by an equation then we can also find the 1 f(x) = x + 4 inverse algebraically. Solution. One way to do this is to write out the equation y = f(x) and solve for x, To find the inverse of f(x) = x + 4 we write y = x + 4 and then solve for getting a function x = g(y): x by subtracting 4 from both sides: y − 4 = x: Once this is done, we bow to custom and swap the letters x and y so that Now that we have solved for x, we swap letters x and y and write x represents the input for our new function and y represents the output. x − 4 = y: So our answer is Thus our final answer will be y = g(x): f −1(x) = x − 4: We will do some examples. We swapped our letter x and y at some point since we are following the tradition that x is the input and y is the output. We could have done this swap of letters at the beginning of the problem and then solved for y. Smith (SHSU) Elementary Functions 2013 15 / 33 Smith (SHSU) Elementary Functions 2013 16 / 33 Function inverses, worked examples Function inverses, worked examples 3 Find the inverse of f(x) = 3x + 5: 2 Find the inverse of f(x) = 3x Solution. To find the inverse of f(x) = 3x we write y = 3x. Solution. Let's swap our letters x and y to announce that we are changing inputs To find the inverse of f(x) = 3x + 5 we first write y = 3x + 5 and then and outputs. (now or later) exchange the letters x and y to indicate that old inputs are now outputs and old outputs are now inputs. So write x = 3y: x = 3y + 5 Then solve for y by dividing both sides by 3: We solve for y by subtracting 5 from both sides and then dividing both sides by 3, to get x=3 = y: x − 5 = y: 3 So our answer is So our answer is −1 f (x) = x=3: x − 5 f −1(x) = : 3 Smith (SHSU) Elementary Functions 2013 17 / 33 Smith (SHSU) Elementary Functions 2013 18 / 33 Function inverses, worked examples Function inverses, worked examples p 3 1 4 f(x) = x + 9 5 f(x) = x + 1 Solutions. p 1 1 To find the inverse of f(x) = x3 + 9 we write Solutions.