Question

Find the inverse of
g(x) = 4x^3 + 5

Answer 1

Answer:

$g^{-1}(x) = \sqrt[3]{\frac{x - 5}{4}}$

Step-by-step explanation:

To find the inverse of a function, you can switch the "x" and "y" variables, then isolate "y", then change it back to function notation with the superscript ⁻¹.

You can't see "y" in the given function, but y = g(x). g(x) is the way you write "y" in function notation. The inverse function notation is g⁻¹(x).

$g(x) = 4x^{3} + 5$  

$y = 4x^{3} + 5$   Change "g(x)" to "y"

$x = 4y^{3} + 5$   Switch the "x" and "y" variables

$x - 5 = 4y^{3} + 5 - 5$   Subtract 5 from both sides to start isolating "y"

$x - 5 = 4y^{3}$   5 - 5 cancels out on right side

$\frac{x - 5}{4} = 4y^{3}/4$   Divide both sides by 4

$\frac{x - 5}{4} = y^{3}$   4y³/4 cancels out the 4 on the right side

$\sqrt[3]{\frac{x - 5}{4}}= \sqrt[3]{y^{3}}$   Cube root both sides

$\sqrt[3]{\frac{x - 5}{4}} = y$   ∛y³ cancels out the ³ on the right side

$y = \sqrt[3]{\frac{x - 5}{4}}$   Variable on left side is standard formatting

$g^{-1}(x) = \sqrt[3]{\frac{x - 5}{4}}$   Change the inverse function notation g⁻¹(x)

The inverse of g(x) = 4x³ + 5 is g⁻¹(x) = ∛[(x-5)/4].