# use the function f(x)=1/8x-3 and g(x)x^3 to find th indicated value or function.  (g^-1 composition f^-1)(-3)?this question is my book but i don't understand what it's asking for. Do I need to...

use the function f(x)=1/8x-3 and g(x)x^3 to find th indicated value or function.  (g^-1 composition f^-1)(-3)?

this question is my book but i don't understand what it's asking for. Do I need to find the inverse function amd the plug it in and then find the composition of it.

neela | High School Teacher | (Level 3) Valedictorian

Posted on

f(x) = 1/(8x-3)

g(x) = x^3.

To find (  g^-1 composition f^-1  )(-3)

Solution:

The notation ^-1 stands for the inverse function.

First let us find the inverse function of f(x) Or y = 1/(8x-3).

Then we find x interms of y by making x  subject. Or by multiplying by 8x-3 and dividing by y.

8x-3 = 1/y

8x= 1/y+3 = (3y+1)/y.

Divide by 8:

x = (3y+1)/8y. Swap x and y.

y = (3x+1)/x is the inverse of f(x) = y = 1/(8x-3).

Or f^-1(x) = (3x+1)/x.

To find the the inverse of g(x) = x^3.

Let y = x^3. Take cube root :

y^(1/3) = x.

x = y^(1/3).

Swap x and y:

y = x^(1/3) is the inverse of g(x) = x^3.

Or g^-1(x) = x^(1/3).

Now g^-1 composition f^-1(3) = g^-1(f^-1(3))

g^-1 (f^-1(x)) =  g^-1 ((3x+1)/3) = {(3x+1)/3}^(1/3).

Therefore , g^-1 (f^-1(-3)) = {3*-3+1)/3}^(1/3) = (-8/3)^(*1/3) =  -2/3^(1/3) = {-2*(3^(2/3)}/3 = -{2*(9)^(1/3)}/3  =  -1.386722549.

Hope this helps.

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