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+5
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if $$f(x)=2x-7,g(x)=\sqrt{3x-2},x\geq 2$$ then

$$(f^{-1}o g^{-1})(5)=?$$

$$(g^{-1}o f^{-1})(-3)=?$$

YehChi  Jan 13, 2015

Best Answer 

 #1
avatar+88898 
+10

Let's find the inverse of both functions

y = 2x - 7  add 7 to both sides                             y= √(3x - 2)   square both sides

y + 7 = 2x     divide both sides by 2                     y^2  = 3x - 2   add 2 to both sides

(y + 7) / 2  = x     exchange x with y                   y^2 + 2 = 3x   divide both sides by 3

(x + 7) / 2  = y  = f-1(x)                                     ( y^2 + 2) / 3  = x     exchange x and y

                                                                         (x^2 + 2 ) / 3  = y = g-1(x)

And g-1(5)   = (25 + 2) / 3 = 9

So   f-1 (g-1(5))  =  f-1 (9) = (9 + 7) / 2  = 16 / 2 = 8

 

And f-1 (-3) = ( -3 + 7) / 2  = 4 / 2 = 2

So g-1 (f-1(-3)) = g-1 (2)  =  (2^2 + 2 ) / 3  = (6) / 3  = 2

 

CPhill  Jan 13, 2015
 #1
avatar+88898 
+10
Best Answer

Let's find the inverse of both functions

y = 2x - 7  add 7 to both sides                             y= √(3x - 2)   square both sides

y + 7 = 2x     divide both sides by 2                     y^2  = 3x - 2   add 2 to both sides

(y + 7) / 2  = x     exchange x with y                   y^2 + 2 = 3x   divide both sides by 3

(x + 7) / 2  = y  = f-1(x)                                     ( y^2 + 2) / 3  = x     exchange x and y

                                                                         (x^2 + 2 ) / 3  = y = g-1(x)

And g-1(5)   = (25 + 2) / 3 = 9

So   f-1 (g-1(5))  =  f-1 (9) = (9 + 7) / 2  = 16 / 2 = 8

 

And f-1 (-3) = ( -3 + 7) / 2  = 4 / 2 = 2

So g-1 (f-1(-3)) = g-1 (2)  =  (2^2 + 2 ) / 3  = (6) / 3  = 2

 

CPhill  Jan 13, 2015
 #2
avatar+93356 
+5

Looks mind goggling. Thanks Chris  

Melody  Jan 14, 2015

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