Beforediscussingself-inverse let mebriefly testimonial the idea of station function. Us think ofadditionand subtraction as opposites and multiplication and department andsquaring and square roots. The an interpretation of inverse functionisa formalization of this idea. Think about the features f(x) = x+ 6and g(x) = x - 6. If you add 6 to x and also thensubtract 6 orif girlfriend subtract and then addyou finish up wherein youstarted,with x. That is,g(f(x)) = f(g(x)) = x.We saythat two attributes f and also g space inverses that each other iff(g(x))= g(f(x)) = x. Provided a duty f(x), if it has actually an inversetheinverse is designated together f-1(x). Other pairs of inverse features are f(x) = 6x and also g(x) = x/6 and also f(x)= x2 and g(x) = sqrt(x).Given an equation favor y = 6x, the inverse have the right to be found bysolving because that x in terms of y. This offers us x = y/6.This is just another method of saying the if us take 6x anddivide through 6 we end where we started, v x.Given a graph of a role f(x), the is straightforward to watch the graph the f-1(x). Listed below is the graphof f(x) = x2 because that x > 0.

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For any type of pair (x, y) on the graph such as (3, 9), the inversefunction, square root of x, bring away 9 come 3, and in general, if (a, b) ison the graph the f(x), (b, a) is on the graph that f-1(x).To check out the point out (b, a), that is only necessary to revolve thegraph 90 degrees and also flip it around so the the graph of square source ofx is:
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In order for this to it is in a function, I got rid of the worths for thenegative x worths of the parabola.
We know that foranyconstant a, the train station of x/a is ax. What aboutthe role for the hyperbola fx(x) = a/x? In this casef(f(x)) = x. A/x is its very own inverse and so is a self-inverse function.Using the an interpretation of symmetry, the duty f(x) = a/x is symmetricwith respect to the operation of inversion.Below is a graph of the function 0.5/x:
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If we rotate the graph the way we rotated the graph above we end upwith the same points. Right here is another method of visualizing the symmetry.

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If we revolve the graph around the heat y=x shown in the illustration we againget the same points. The graph is symmetric about the line y= x. This will be true of any self-inverse function.We know that for a point (a, b) because that f(x) the point (b, a) ison the inverse f-1(x). Since f(x) = f-1(x)for self invese functions, because that any suggest (a, b) top top f(x) thepoint (b, a) must additionally be on f(x). In the picture, two suchpoints A and B have actually been chosen. The midpoint that A and also B is ((a + b)/2,(a + b)/2), i beg your pardon lies on the line y = x. The slopeof the line through A and B is -1. The slope of the line y=x is 1.The product that the slopes is -1, an interpretation that the two linesare perpendicular, so that the line y = x is theperpendicular bisector the segment abdominal muscle and consequently rotating aboutthe line y=x swaps the location of A and B.If us look at the equation y = 0.5/x, that does not show up atfirst to hint at any kind of symmetry. Have the right to you think that a basic wayof composing the equation so that the the contrary is revealed?If we merely multiply both political parties by x, we acquire xy = 0.5. Thesymmetry is currently clear. Swapping x and y offers the sameequation. Since x and also y have actually the same duty in the function,solving for x in terms of y will have actually the same form as the functionexpressing y in regards to x. The is currently clear that the functiony = 10 - x is a self-inverse because it is the exact same as x + y =10. Making use of this building we deserve to now easily generateself-inverse functions. Because that example, we have the right to construct theself-inverse function xy = x + y. This states that because that a givenvalue the x, y is such the multiplying the by x is the exact same as adding itto x. If we fix for y we get y = x/(x-1). Exercise: Verifythat f(x) = x/(x - 1) is a self-inverse function.Soltution:Substituting x/(x - 1) because that x in x/(x - 1), f(f(x)) = x/(x -1)/ (x/(x - 1) - 1) . Simplifying the denominator, x/(x - 1) - 1 = (x - (x - 1))/(x- 1) = 1/(x - 1), so we have x/(x - 1) / (11/ (x - 1)) = (x / (x - 1))(x - 1) = x and so f(x) is self-inverse.Exercise:Verify the f(x) = x/(x - 1) satisfies xf(x) = x + f(x)Solution:xf(x) = x2/(x - 1). X + f(x) = x + (x/(x - 1)) = (x(x - 1) + x)/ (x - 1) = (x2- x + x)/(x - 1) = x2/(x - 1).
Now look at the equation y = x2/(x - 1). Over there is a discontinuity at x = 1, so the function dividesinto 2 pieces because that x 1. Let"s look at theportion because that x > 1. It must be clear that for x near 1 and forlarge worths of x, the duty goes to infinity, so we deserve to supposethat that takes on a minimum value in between.I come up v this function by taking theself-inversefunction x/(x - 1) over and multiply it by x. The is,f(x) = x(x/(x-1)). We can use this construction as in the instance of theparabola to display that f(x/(x - 1)) =(x/(x - 1))x =f(x). If we assume that the function for x 2 - x = x, x2 - 2x = 0, x(x - 2) = 0. Because that x> 1, the just solution is x = 2, because that which f(x) = 4.Looking at the graph of the function it deserve to be viewed that wewere correct in assuming that the duty takes ~ above a minimum at onlyone place and that the minimum does indeed occur for x = 2. The othersolution for x(x - 2) =0, x = 0, is a maximum because that the part of thegraph where x
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