Find a formula for the inverse of the function. g(x) = x^2 - 2x, x >= 1

Question

Please show steps to get the answer.

Answer 1

g(x) has no inverse function, until you take account of the restriction x>1

y=x^2-2x

switch x and y

x=y^2 -2x

solve for the new y

x+1 = y^2-2x+1

x+1 = (y-1)^2

y-1 =+/-sqr(x+1)

y = 1+/-sqr(x+1)

but ignore the positive square root, due to the restriction x>1 for g(x)

y =g^-1(x) = 1+sqr(x+1)

it may help to draw a rough sketch.

g(x) is the right half an upward opening parabola if x>1

g^-1(x) is the bottom half a rightward opening parabola, if x>1 for g(x)

g^-1(g(x)) = g^-1(x^2-2x+1)

= 1+sqr(x-1)^2

= 1+x-1= x

g(g^-1(x)) also = x

the composite function of inverse functions of x = x

Answer 2