Question

How to find the inverse function for a quadratic equation?

This is the equation f(x)= x^2+6 x+14, x∈(−∞,-3]. I have tried every method I can think of and still can not figure out the inverse function.

Answer 1

`"See explanation"`

Explanation:

`y = f(x) = x^2 + 6x + 14`
`"There are two methods that one can follow."`
`"1) Completing the square : "`
`y = (x+3)^2 + 5`
`=> pm sqrt(y - 5) = x + 3`
`=> x = -3 pm sqrt(y - 5)`
`=> y = -3 pm sqrt(x - 5)" is the inverse function."`
`"For "x <= -3" we take the solution with - sign."`
`=> y = -3 - sqrt(x-5)`
`"2) Substituting "x = z + p", with " p" a constant number"`
`y = (z + p)^2 + 6(z + p) + 14`
`= z^2 + (2p + 6) z + p^2+6p+14`
`"Now choose "p" so that "2p + 6 = 0 => p = -3.`
`=> y = z^2 + 5`
`=> z = pm sqrt(y - 5)`
`=> x = -3 pm sqrt(y - 5)`