You are watching: Functions that are their own inverse
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:
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:
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