Math 1330 – 1.5, Inverse FunctionsTwo functions are inverses of one another provided successive composition, one with the other, results in “x”.In symbols: If f and g are functions and inverses, thenf(g(x)) = x and g(f(x)) = x. We then denote ).x(g)x(for)x(f)x(g11 So, we’ll be looking at Big Picture facts about all inverse functions first, and then we’ll bear down on the details of individual functions and their exact and specific inverses.Now, there’s four basic truths other than this definition:T1 If (a, b) is on f(x), then (b, a) is on the inverse function.T2 If g is the graph of f(x) then the graph of f inverse is g reflected over the line y = x. You are swapping axes.T3 To find the functional form of the inverse function, swap x and y and solve for y.[Note that in all 3 cases, you swap x and y to find out information about the inverse function.]T4 To solve for an x that is an expression in the argument of a function, take the inverse function of both sides of the equation.(note: this is College Algebra, Chapter 3, Section 7 – check out the streaming videos for both our text and the College Algebra course if you need more than a quick review)1T1 If (a, b) is on f(x), then (b, a) is on the inverse function.Let’s look again at this statement. Let’s take the point (2, 3) on a pair of axes.If I swap the x and y…which coordinate is now on the horizontal axis?Now let’s look at how this works for inverse functions:Here’s a list of points on a function Let’s make a list of the points that The function has domain ),0[ are on )x(f1( 0, 0)41,21(1, 1)(2, 4)(3, 9)(4, 16)…What function is this…and what is it’s inverse?Now let’s look at some information…and I’m not going tell you anything but f and g are inverse functions to one another:On a problem solving note:If I tell you that g(x) is the inverse function for f(x) and thatf(3) = 2, f(2) = 5, f(6) = 122I can then ask, what is g(2)?Organize the facts and answer the question:Here’s another one: h and m are inverses.h(0.5) = 5 h(5) = 9 h(9) = 3 what is m(5)?Organize, answerYou have to be systematic and organized so that swapping x and y works in an orderly fashion.3T2 If g is the graph of f(x) then the graph of f inverse is g reflected over the line y = x. You are swapping axes.Below is the graph of f(x). (It’s a piece-wise defined function, right? Restricted domain and range.) Show the graph of it’s inverse function:What are the domain and range for each function?Here’s the graph of another function. Show the graph of it’s inverse function:4( 3, 1)(4, 2)(6, 4)T3 To find the functional form of the inverse function, swap x and y and solve for y.Suppose I give you a function: f(x) = 3x + 5Swap x and y x = 3y + 5solve for yTHIS y is 1fOn your own: check itxffff11NOTE:The domain and range of the original function are swapped for the inverse function.If you haveOriginal Function Inverse functionDomain is [0, �) Domain is (�, 5]Range is (�, 5] Range is [0, �)You’ll really see this in the last example on page 9.5On your own do these problems using the preceding functions. See a CASA tutor or come to my conference hours to get your work checked.f(1/3) =)3/1(ff1)6(f1)6(ff1Consistently, you see, you return to your starting place of “x”…that’s what we mean when we have the definitionxffff116Now we come to an interesting question – do all functions have inverse functions? The answer is no. There’s a whole class of functions who are guaranteed to have a related inverse function. Some of the other kinds of functions have inverse relationships that are not functions:Here’s one with no inverse function. Let’s graph it and then graph what ought to be it’s inverse function and see what goes wrong:eg2x)x(f The inverse we care about is the inverse FUNCTION…ie, it passes the vlt just like the original function does.The functions that have related inverses that are functions are called 1:1 functions – they pass the VLT for the “function” part and they pass the horizontal line test for the “1:1” part…thus ensuring the inverse, the swapped x and y graph, is a function, too.Note that a horizontal line for f(x) is a vertical line for )x(f1 when you swap x and y axes. So we do the VLT to test to see if it’s a function and the HLT to see if it has an inverse.7Let’s look at some samples of graphs and pick out which ones are 1:1. If a function is 1:1, then it always has an inverse function…so solve for the inverse function if there is one.8Let’s do one whole problem: here’s a function1x1x2)x(fIt IS one-to-oneWhat does f inverse look like on a graph? BE REALLY CAREFUL WITH THOSE ASYMPTOTES!On your own:9VA x = HA y = x intercepty interceptdomainrangeVAHAx intercepty interceptcritical pointsdomainrangeWhat is the functional form of the inverse? 1x1x2)x(fSwap x and ySolve for yWhat are the domains and ranges of bothhint: the original function domain is all Real numbers except 2.10T4 To solve for an x that is an expression in the argument of a function, take the inverse function of both sides of the equation.One last use of inverse functions – and this is something you’ve done before, but you probably haven’t thought about this way.If you’re going to solve52x3 what do you do?And are you multiplying the LHS and the RHS by exactly the same number?You’re actually taking a function of the LHS and the same function of the RHS…which function is this?For a while now you’ve been taking a function of both sides when the functions weref(x) = x and 21x)x(f .So suppose I don’t tell you which function you’ve actually got but I do tell you one fact…as inf(3x+2) = 5 and 25)5(f1. Can you solve for x?How about3)x2(f1 and f(3) =  6. Solve for x.And a last one:1f (5 x) 14 f (14) 20-- = =11You need to add the technique of taking a function of both sides CONSCIOUSLY to