Question

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

Answer 1

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

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

`(x - 5)/(4) = 4y^(3)/4` Divide both sides by 4

`(x - 5)/(4) = y^(3)` 4y³/4 cancels out the 4 on the right side

`\sqrt[3]{(x - 5)/(4)}= \sqrt[3]{y^(3)}` Cube root both sides

`\sqrt[3]{(x - 5)/(4)} = y` ∛y³ cancels out the ³ on the right side

`y = \sqrt[3]{(x - 5)/(4)}` Variable on left side is standard formatting

`g^(-1)(x) = \sqrt[3]{(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].