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Hi , Show that the inverse of a linear fraction function is always a linear fraction function (except where it is not defined).
Hi ,
Show that the inverse of a linear fraction function is always a linear fraction function (except where it is not defined).
Let, f(x) = (a+bx)/(c+dx) be the said linear fraction function. Let at some x it attains value y, so, (a+bx)/(c+dx) = y => a + bx - cy - dxy = 0 => a - cy + x (b - dy) = 0 => x = (cy-a)/(b-dy). Which is again a linear fraction function defined in R except at x = -c/d and y = b/d and inverse of the given function is, y = (cx-a)/(b-dx).
Let, f(x) = (a+bx)/(c+dx) be the said linear fraction function.
Let at some x it attains value y, so,
(a+bx)/(c+dx) = y
=> a + bx - cy - dxy = 0
=> a - cy + x (b - dy) = 0
=> x = (cy-a)/(b-dy).
Which is again a linear fraction function defined in R except
at x = -c/d and y = b/d
and inverse of the given function is, y = (cx-a)/(b-dx).
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