f sends 1 to 2, so f 1 sends 2 back to 1. f sends 2 to 4, so f 1 sends 4 back to 2. f 1 = { (2,1), (4,2), (6,3), (1,4), (3,5), (5,6) }.

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1 .3 Inverse Functions an their Derivatives In this unit you ll review inverse unctions, how to in them, an how the graphs o unctions an their inverses are relate geometrically. Not all unctions can be unone, but when they can be, the inverse unction unoes whatever the unction i. I the unction ties a knot, then the inverse unction unties the knot. Then we ll apply the chain rule to in the erivatives o inverse unctions. Inverse Functions First let be the unction = { (,), (,), (3,6), (,), (5,3), (6,5) } with arrow iagram: Since the inverse unction unoes what i, it s easy to in the arrow iagram or this inverse unction; just raw in the arrows that clean up the mess create, the arrow iagram that unoes the unction : This iagram was constructe by thinking: Thus, sens to, so sens back to. sens to, so sens back to. Etc = { (,), (,), (6,3), (,), (3,5), (5,6) }. Notice that the points in are simply the points in, but with all their xy-coorinates switche! For example, in place o the point (,), the inverse unction has the point (,). Below is a graph o unction, along with its inverse unction,. Plotting in blue, an its inverse in re, we get the iagram to the right: The iagonal line y=x is graphe in yellow; notice the symmetry o the graphe points about that line.

2 Important concept! Switching the xy-coorinates o any point relects that point through the iagonal line y=x, as illustrate in the iagram: Since all the points o the graph o the inverse unction are points o the original unction with the xycoorinates switche, it ollows that the graph o the inverse unction is the graph o the unction relecte through the line y=x, as we saw to be the case in the above example. Our simple example o a unction an its inverse worke because not only i our original unction pass the vertical line test (vertical lines cross the graph o ANY unction in at most one place), but it also passes the horizontal line test (horizontal lines cross the graph o the unction in at most one place). In such cases the unction is sai to be one-to-one, which means each input to the unction (those are the x-coorinates in the points that make up the graph o the unction) is paire with a unique output rom the unction (those are the y-coorinates in the points that make up the graph o the unction), an also each output rom the unction has a unique input that generates it. IMPORTANT! The inverse o a unction is not its reciprocal, in general: The notation is unortunately ambiguous, but whenever you see (x), this means the inverse unction an not the reciprocal o the unction. Most unctions we nee eal with are eine in terms o equations, usually where y is expresse algebraically as a unction o x. Fining the inverse unction algebraically also involves switching the x an y coorinates. For example, to in the inverse unction o the unction = 5, we start by switching the x s an y s: y = 5 x = 3y 5 Then solve the resulting switche equation or y: That s the inverse unction, x + 5 =. 3 x + 5 x + 5 = 3y = y. 3

3 Here are both unctions plotte on the same graph, along with the line y=x. NOTE: To in an inverse unction when the equation or the unction expresses y as a unction o x, we ve irst switche the x s an y s in the equation, an then solve or y. But you can also o this in the reverse orer, irst solving or x as a unction o y an then switching the x s an y s in the equation. Examples:. Let y = +. Fin y in terms o x an y in terms o y. We are given y as a unction o x: y +. y y 3 ( x+ ) = 3 ( x+ ) = 3 ( x+ ) ( + ) = 3 ( x+ ) 3 + y 6 ( + ) Solving or x as a unction o y, we get: y y = y( + ) = y+ y = y = y x= + 3y ' y = =. 3y That last line is the erivation o the inverse unction, x ( y) Dierentiating with respect to y, we get: ( 3y) ( y) ( y) ( 3y) y y y 3y 3 3y 6+ 3y x y y 3y 3y 3y 6 = y y 9y y 3y y ( y) ( 3y)

4 Notice that the equation ( y) y = eines the unction 3y x =. It s the same unction either way! just as well as the equation Plotting an its inverse = in blue, + x = in re, we get the iagram to the right: The iagonal line y=x is graphe in yellow; notice the symmetry o the graphe unctions about that line.. Let y ax + b, an let x = ( y). Fin y in terms o x an in terms o y. cx + y ( cx + ) ( ax + b) ( ax + b) ( cx + ax + b y ax + b ) y = cx + cx + cx + ( cx + ) a ( ax + b) c acx + a acx bc a bc ( cx + ) ( cx + ) ( cx + ) = y a bc = cx ( + ) Solving or x as a unction o y (an thus ining the inverse unction), we get: ax + b y = y ( cx + ) = ax + b cxy + y = ax + b cxy ax = b y cx + Dierentiating with respect to y, we get: b y cy a ( cy a) x = b y x ( y) ( cy a) ( b y) ( b y) ( cy a) b y y y cy a b y c x = y y cy a cy a cy a cy a bc cy a bc = y + + ( cy a) y ( cy a) ( ) ( ) ( ) ( y )

5 The Derivative o an Inverse Function To in the erivative o an inverse unction,, we begin with the algebraic equation that escribes the act that the unction unoes whatever the unction oes to x (since these unctions are inverses o each other), an then we ierentiate both sies o that equation: ( ) ( ) x = x x x Now we must invoke the chain rule, I we let y = an x ( y) ( ) = x = ( g x ) = '( g ) g', to get '( ) = ( x ) = ' x ( ) =, then the above equation can be expresse more concisely: = y y In the next section we ll use this technique to erive ormulas or the inverse trigonometric unctions. Derivative Examples: 3. Let y = tan x. Fin y at π x =. π π At x =, we have y = tan =, so we want to in the erivative o the inverse unction o y = tan x, at the point y =. This is the reciprocal o the erivative y evaluate at π x =. y π But π = x π = π = = tan sec sec =. x = x = x =

6 An so, y y = y π x =. Let y = x sin x. Fin y at x = 0. At x = 0, we have y = 0 sin0= 0, so we want to in the erivative o the inverse unction o y y = xsin x, at the point y = 0. This is the reciprocal o the erivative evaluate at 0 x =. y But = ( x sin x ) = ( cos x ) = cos 0 = = 0. x = 0 x = 0 x = 0 The inverse unction s erivative is then the reciprocal o 0, which is uneine. At any points on the graph o y = x with horizontal tangent lines, the corresponing points on the graph o y = x points. Plotting an its inverse have vertical tangent lines. The erivative o the inverse unction isn t eine at such y = = x sin x in blue, y x = in re, we get the iagram to the right: The iagonal line y=x is graphe in yellow; notice the symmetry o the graphe unctions about that line. Also notice that at (0,0), has a horizontal tangent (the x-axis) while has a vertical tangent (the y-axis).