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.

1 Answer
MattyMatty
Mar 8, 2018

#"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)#

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