Question

Find the inverse f^-1(x) of the following function:
f(x)=x^(2/3)+3x^(1/3)-2

Answer 1

\[f(x)=x^{2/3}+3x ^{1/3}-2\]

Answer 2

\[ \large
f(x)=x^{2/3}+3x ^{1/3}-2 \\ \large
f(x) = (x^{1/3})^2 + 3x^{1/3} + \frac 94 - \frac 94 - 2 \\ \large
y = (x^{1/3} + \frac 32)^2 - \frac{17}{4} \\
\text{To find inverse, switch x and y and solve for y:} \\ \large
x = (y^{1/3} + \frac 32)^2 - \frac{17}{4} \\ \large
x + \frac{17}{4} = (y^{1/3} + \frac 32)^2 \\ \large
\sqrt{x + \frac{17}{4} } = (y^{1/3} + \frac 32) \\ \large
y^{1/3} = \left (\sqrt{x + \frac{17}{4} } - \frac 32 \right) \\ \large
y = \left (\sqrt{x + \frac{17}{4} } - \frac 32 \right)^3 \\ \large
f^{-1}(x) = \left (\sqrt{x + \frac{17}{4} } - \frac 32 \right)^3
\]

Answer 3

or \[ \large
f^{-1}(x) = \left (\frac{\sqrt{4x + 17}-3}{2} \right)^3
\]

Answer 4

Inverse is valid for \(\large x \ge -\frac{17}{4}\)

Answer 5

Thanks so much!