[tex]g( \: f(x) \: ) = x[/tex]
And
[tex]f( \: g(x) \: ) = x[/tex]_________________________________
So :
[tex]g(x) = {f}^{ - 1}(x) [/tex]
And
[tex]f(x) = {g}^{ - 1}(x) [/tex]
_________________________________
So to find g(x) , we must find the inverse of f(x) .
Let's do it .....
[tex]f(x) = 8 {x}^{3} + 1 [/tex]
[tex]y = 8 {x}^{3} + 1 [/tex]
Subtract the sides of the equation minus 1
[tex]y - 1 = 8 {x}^{3} [/tex]
Divided the sides of the equation by 8
[tex] \frac{y - 1}{8} = {x}^{3} \\ [/tex]
From the sides of the equation, we take the radical with interval 3
[tex] \sqrt[3]{ \frac{y - 1}{8} } = x \\ [/tex]
[tex] \sqrt[3]{ \frac{1}{8}(y - 1) } = x \\ [/tex]
[tex] \frac{1}{2} \sqrt[3]{y - 1} = x \\ [/tex]
[tex] \frac{ \sqrt[3]{y - 1} }{2} = x \\ [/tex]
So ;
[tex] {f}^{ - 1}(x) = \frac{ \sqrt[3]{x - 1} }{2} \\ [/tex]
Now we find g(x) which is :
[tex]g(x) = \frac{ \sqrt[3]{x - 1} }{2} \\ [/tex]
_________________________________
And we're done.
Thanks for watching buddy good luck.
♥️♥️♥️♥️♥️
If f(g(x)) = x and g(f(x)) = x , than [tex]\rm g(x) = \sqrt[3]{\dfrac{x-1}{8}}[/tex] and this can be calculated by finding the inverse of the given function f(x).
Given :
Given that f(g(x)) = x it means that:
[tex]\rm g(x) = f^{-1}(x)[/tex]
It is also given that g(f(x)) = x that means:
[tex]\rm f(x) = g^{-1}(x)[/tex]
Now, the inverse of f(x) is given by the following calculation:
[tex]\rm f(x) = 8x^3+1[/tex]
Now, replace f(x) with y in the above equation.
[tex]y = 8x^3+1[/tex]
Now, replace y with x and x with y and evaluate 'y'.
[tex]x = 8y^3 +1[/tex]
[tex]\sqrt[3]{\dfrac{x-1}{8}}=y[/tex]
Now, replace y with [tex]\rm f^{-1}(x)[/tex] in the above equation.
[tex]\rm f^{-1}(x) = \sqrt[3]{\dfrac{x-1}{8}}[/tex]
[tex]\rm f^{-1}(x) = \sqrt[3]{\dfrac{x-1}{8}} = g(x)[/tex]
For more information, refer to the link given below:
https://brainly.com/question/13715269
