Respuesta :

alainvatel
Replace f(x)fx with yy.y=6x3−8y=6x3-8Interchange the variables.x=6y3−8x=6y3-8Solve for yy.
y=3√36(x+8)6y=36x+836Solve for yy and replace with f−1(x)f-1x.
f−1(x)=3√36(x+8)6

Answer:

[tex] f^{-1}(x) = \sqrt[3]{\frac{x+8}{6}}[/tex]

Step-by-step explanation:

Given function,

[tex]f(x)=6x^3-8[/tex]

Replace f(x) by y,

[tex]y=6x^3-8[/tex]

Switch x and y,

[tex]x=6y^3-8[/tex]

Isolate y in the left side,

[tex]-6y^3=-x-8[/tex]

[tex]6y^3=x+8[/tex]

[tex]y^3=\frac{x+8}{6}[/tex]

[tex]y=\sqrt[3]{\frac{x+8}{6}}[/tex]

Replace y by [tex]f^{-1}(x)[/tex]

[tex]\implies f^{-1}(x) = \sqrt[3]{\frac{x+8}{6}}[/tex]

Which is the required inverse of the given function.