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# invers function?

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g(x)=(3/(x^2)-16)+(2/x+4) ..

a.) g^1(x) (g inverse x =..?)

b.) g^1(5)= ?

I'm Indonesian BTW :D

Mar 15, 2018

#1
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g(x)   = y

y  =       3                    2 (x - 4)

_______    +      _________

x^2 - 16             (x + 4)(x - 4)

y  =    3 + 2x - 8

__________

x^2  - 16

y  =  2x -  5

_______

x^2  - 16

y (x^2 - 16)  = 2x - 5

yx^2 - 16y  = 2x - 5

16y  =  yx^2 - 2x + 5

16y  = y (x^2 - (2/y)x + 5/y)

16 =  x^2  - (2/y)x + 5/y      complete the square on x

16  = x^2 - (2/y)x + 5/y + 1/y^2 - 1/y^2

Add  1/y^2  to both sides....subtract 5/y from both sides

16 + 1/y^2 -5/y   = x^2 -(2/y)x + 1/y^2      factor the right side

16 + 1/y^2 - 5/y  = (x - 1/y)^2

1/y^2 - 5/y + 16  =  (x - 1/y)^2      get a common denominator on the left

[16y^2 -5y + 1] / y^2  =  (x - 1/y)^2      take both roots

±√[ (16y^2 -5y + 1) / y^2 ]  =  x - 1/y      add 1/y to both sides

1/y ±√  (16y^2 -5y + 1) / y

[ 1 ±√  (16y^2 -5y + 1)  ] / y  = x       "swap" x and y

[ 1 ±√ [ 16x^2 - 5x + 1) ] / x  =  y  =  g-1(x)

Thus....this is the inverse.....but it is not one-to-one.....we have two values for g-1(5)

g-1 (5)  =   ( 1 + √ [ 16(5)^2 - 5(5) + 1 ] )  / 5   = ( 1 + √ 376 ) /  5  ≈ 4.078

g-1(5) =  ( 1 - √ [ 16(5)^2 - 5(5) + 1 ] )  / 5   = ( 1 - √ 376 ) /  5  ≈ -3.678

So   g-1 (5)   =   ≈4.078    and   ≈ -3.678

Mar 15, 2018