Question

Find the inverse of the function f(x)= 6x^3-8

Answer 1

Hello, please consider the following.

$(\forall x \in \mathbb{R}) \\ \\ x=(f^{-1}of)(x)\\\\=(fof^{-1})(x)\\\\=f(f^{-1}(x))\\\\=6\left( f^{-1}(x) \right) ^3-8 = x\\\\\text{We need to express }f^{-1}(x) \text{ in function of x.}\\\\\text{Let's do it!}$

$6\left( f^{-1}(x) \right) ^3-8 = x\\\\\left( f^{-1}(x) \right) ^3 =\dfrac{x+8}{6}\\\\\Large \boxed{\sf \bf \ \ f^{-1}(x)=\sqrt[3]{\dfrac{x+8}{6}}\ \ }$

Thank you

Answer 2

To find the inverse, interchange the variables and solve for
y
y
.
f
−
1
(
x
)
=
√
2
(
x
+
8
)
6
,
−
√
2
(
x
+
8
)
6