There are several ways of finding an inverse function but I prefer to use the following which relies on two definitions (i) the definition of an inverse and (ii) the definition of the function in question. So if f(x) = 6 - (6/x^2) then suppose that f^-1(x) = k then by definition of an inverse f(k) = x. But by definition of this particular function, f(x) = 6 - (6/x^2) so therefore f(k) = 6 - (6/k^2) and f(k) = x so that means that x = 6 - (6/k^2) and we make k the subject of the formula, or solve the equation for k. x - 6 = -6/k^2 or (x - 6)/1 = - 6/k^2 so since we now have two fractions, invert both and they will still be equal so 1/(x - 6) = (k^2)/6 or (k^2)/6 = 1/(x - 6) so now multiply both sides by 6 and k^2 = 6/(x - 6) and now take the square root so that k = +/- sqrt[6/(x - 6)]. However, if there is to be an inverse i.e. an inverse function, there must be only one answer, so at this point you have to decide whether you are going to use the positive root or the negative one.
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f(x) = 6 - (6/x²))
=> f(f⁻¹(x)) = 6 - (6/(f⁻¹(x))²))
=> x= 6 - (6/(f⁻¹(x))²))
=> 6/(f⁻¹(x))²) = 6 - x
=> 6/(6 - x) = (f⁻¹(x))²
=> f⁻¹(x) = √(6/(6 - x)) and f⁻¹(x) = -√(6/(6 - x))
There are several ways of finding an inverse function but I prefer to use the following which relies on two definitions (i) the definition of an inverse and (ii) the definition of the function in question. So if f(x) = 6 - (6/x^2) then suppose that f^-1(x) = k then by definition of an inverse f(k) = x. But by definition of this particular function, f(x) = 6 - (6/x^2) so therefore f(k) = 6 - (6/k^2) and f(k) = x so that means that x = 6 - (6/k^2) and we make k the subject of the formula, or solve the equation for k. x - 6 = -6/k^2 or (x - 6)/1 = - 6/k^2 so since we now have two fractions, invert both and they will still be equal so 1/(x - 6) = (k^2)/6 or (k^2)/6 = 1/(x - 6) so now multiply both sides by 6 and k^2 = 6/(x - 6) and now take the square root so that k = +/- sqrt[6/(x - 6)]. However, if there is to be an inverse i.e. an inverse function, there must be only one answer, so at this point you have to decide whether you are going to use the positive root or the negative one.