We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
209
1
avatar

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
avatar+100571 
+1

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

 

 

 

cool cool cool

 Mar 15, 2018

15 Online Users

avatar