Final answer:
To find the inverse of a cube root function, swap x and y in the original function and solve for the new y. This will result in the inverse function being the cube of the variable. Cube roots and cubes are inverse operations, and calculators can greatly aid in these calculations.
Step-by-step explanation:
To find the inverse of a cube root function, you essentially want to reverse the operation. Let’s say you have a cube root function f(x) = ∛[3]{x}.
The inverse of this function, denoted as f⁻¹(x), would undo the cube root operation by cubing the variable. For example, if f(x) = y, then x = f⁻¹(y) or x = y³. To derive the inverse, swap the x and y in the original equation and then solve for the new y, that will give you the inverse. This can be demonstrated as follows:
- Start with y = ∛[3]{x}.
- Swap x and y to get x = ∛[3]{y}.
- To find y, cube both sides: x³ = y, so y = x³.
- Therefore, the inverse function is f⁻¹(x) = x³.
Remember that the inverse of a function must pass the horizontal line test, meaning that each y-value has a unique x-value. It is also important to note that the cube and cube root are inverse operations much like how the natural log and exponential function undo each other.
Whenever calculating such functions, using a calculator that supports these operations can be extremely helpful. Make sure to familiarize yourself with the calculator functions or seek assistance if needed.