Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f(x) = quantity x minus nine divided by quantity x plus five. and g(x) = quanti

Question

 Cant figure this thing out

Answer 1

 

y = f(x) = (x-9)(x+5) <--- the given function

 

x = (y-9)/(y+5) <--- swaps x and y

 

x(y+5) = y-9

 

xy + 5x = y - 9

 

xy - y = -5x-9

 

y(x-1) = (-5x - 9)

 

y = G(x) = (-5x - 9)/(x-1) is the inverse

 

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f(G(x)) shall be computed as top/bottom

 

top : (-5x-9)/(x-1) - 9 =   [(-5x-9) - 9(x-1) ]/(x-1) = [ -5x - 9 - 9x + 9]/(x-1) = -14x/(x-1)

bottom (-5x-9)/(x-1) + 5 = [ (-5x - 9) + 5(x-1) ]/(x-1) = [ -5x - 9 + 5x - 5]/(x-1) = -14/(x-1)

The quotient top divided by bottom is in fact x which shows they are identities

 

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G(f(x)) is computed in the same way:

 

top :  -5 [  (x-9)/(x+5) - 9] = (-5x + 45)/(x+5) - 9 = [(-5x + 45) - 9(x+5)]/(x+5) = -14x/(x+5)

bottom:   (x-9)/(x+5) - 1 = [(x-9) - (x+5)]/(x+5) = -14/(x+5)

the quotient , top divided by bottom is again x