Does #z(x) = sqrt(x^2-5x-6)# have an inverse function?

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Answer 1 · 2018-02-08T22:29:52

No...

Given:

#z(x) = sqrt(x^2-5x-6)#

Let #y=z(x) = sqrt(x^2-5x-6)#

Square both ends to get:

#y^2 = x^2-5x-6#

Subtract #y^2# from both ends to get:

#0 = x^2-5x-(y^2+6)#

#color(white)(0) = (x-5/2)^2-(y^2+49/4)#

#color(white)(0) = (x-5/2-sqrt(y^2+49/4))(x-5/2+sqrt(y^2+49/4))#

So:

#x = 5/2+-sqrt(y^2+49/4)#

This does not define an inverse function, but a relation, since there is more than one value of #x# for each value of #y#.