Question

Consider the function f(x) = 8x^3 +1. If f(g(x)) = x and g(f(x)) = x , find g(x) ?

Answer 1

$g( \: f(x) \: ) = x$

And

$f( \: g(x) \: ) = x$_________________________________

So :

$g(x) = {f}^{ - 1}(x)$

And

$f(x) = {g}^{ - 1}(x)$

_________________________________

So to find g(x) , we must find the inverse of f(x) .

Let's do it .....

$f(x) = 8 {x}^{3} + 1$

$y = 8 {x}^{3} + 1$

Subtract the sides of the equation minus 1

$y - 1 = 8 {x}^{3}$

Divided the sides of the equation by 8

$\frac{y - 1}{8} = {x}^{3}$

From the sides of the equation, we take the radical with interval 3

$\sqrt[3]{ \frac{y - 1}{8} } = x$

$\sqrt[3]{ \frac{1}{8}(y - 1) } = x$

$\frac{1}{2} \sqrt[3]{y - 1} = x$

$\frac{ \sqrt[3]{y - 1} }{2} = x$

So ;

${f}^{ - 1}(x) = \frac{ \sqrt[3]{x - 1} }{2}$

Now we find g(x) which is :

$g(x) = \frac{ \sqrt[3]{x - 1} }{2}$

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And we're done.

Thanks for watching buddy good luck.

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