SOLUTION: What is the inverse of the function f(x)=64x^3? f^-1(x)=_________

Algebra ->  Inverses -> SOLUTION: What is the inverse of the function f(x)=64x^3? f^-1(x)=_________      Log On


   

Question 258498: What is the inverse of the function f(x)=64x^3?
f^-1(x)=_________
Found 2 solutions by edjones, jsmallt9:
Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
f(x)=64x^3
y=64x^3
x=64y^3
y^3=x/64
y=root%283%2Cx%29%2F4
f%5E-1%28x%29=root%283%2Cx%29%2F4

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
To find the inverse of a function:
  1. Repalce the function notation with "y"
  2. Swap the x's and y's in your equation. This swapping of the x and y changes to equation from the equation of the function to the equation of the inverse.
  3. Solve the inverse equation for y, if possible.
  4. If successful in solving the inverse for y, then the inverse is a function, too. Replace the y with the notation for inverse functions.

Let's see how this works:
f%28x%29+=+64x%5E3
1. Replace the function notation with "y":
y+=+64x%5E3
2. Swap the x's and y's:
x+=+64y%5E3
This is the equation for the inverse relation.
3. Solve the inverse for y, if possible. We'll start by dividing both sides by 64:
x%2F64+=+y%5E3
Next we'll find the cube root of each side:
root%283%2C+x%2F64%29+=+root%283%2C+y%5E3%29
Using the property of radicals, root%28a%2C+p%2Fq%29+=+root%28a%2C+p%29%2Froot%28a%2C+q%29, on the left side and simplifying the cube root on the right side we get:
root%283%2C+x%29%2Froot%283%2C+64%29+=+y
Since root%283%2C+64%29+=+4, this simplifies to:
root%283%2C+x%29%2F4+=+y
4. If successful in solving for y, then replace the y with the notation for inverse functions:
root%283%2C+x%29%2F4+=+f%5E%28-1%29%28x%29 or %281%2F4%29root%283%2C+x%29+=+f%5E%28-1%29%28x%29

(NOTE: Algebra.com's software mistakenly puts a multiplication symbol in the inverse function notation. This is a mistake. There is no multiplication involved.)