SOLUTION: find the inverse of f(x)= 1/3x^3+1 is it one-to-one, why or why not?

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Question 229892: find the inverse of f(x)= 1/3x^3+1 is it one-to-one, why or why not?
Found 2 solutions by stanbon, jsmallt9:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
find the inverse of f(x)= (1/3)x^3+1 is it one-to-one, why or why not?
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Inverse:
Interchange x and y to get:
x = (1/3)y^3 + 1
(1/3)y^3 = x-1
y^3 = 3(x-1)
y = [3x-3]^(1/3)
--------------------------
One-to-One?
Graph to see:
graph%28400%2C300%2C-20%2C20%2C-4%2C4%2C%281%2F3%29x%5E3+%2B+1%29
===================================================
f passes the vertical line test and the horizontal line test
so it is one-to-one
Cheers,
Stan H.

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
To find the inverse of a function:
  1. If the function is written in function notation replace the f(x) (or whatever) with a y.
  2. Rewrite the equation with x in the place of y and y in the place of x. This changes the equation from the equation of the function to the equation of the inverse relation.
  3. Solve the inverse equation for y, if you can.
  4. If you are able to solve for y and express y as equal to a single expression (without use of +-)
    • The inverse is a function
    • The original function was one-to-one. This is so because
      • The original function, as all function do, maps each x to a single y.
        • The inverse which we now have shown to be a function, also maps each of its x's to a single y.
      • Since the inverse is the function with its x's and y's swapped, an inverse which is a function is mapping each of the y's of the original function to a single x of the original function.
      • So each x is mapped to a single y and each y is mapped to a single x. This is what one-to-one means.

Let's try this on your function:
f%28x%29=+%281%2F3%29x%5E3%2B1
1. Replace f(x) with y:
y=+%281%2F3%29x%5E3%2B1
2. Swap the x's and y's. This creates the equation for the inverse relation:
x=+%281%2F3%29y%5E3%2B1
3. Solve for y if you can:
Subtract 1 from each side:
x-1=%281%2F3%29y%5E3
Multiply both sides by 3:
3x-3+=+y%5E3
Find the cube root of each side. (Note: If this was an even-numbered root instead of an odd-numbered root, we would have to use a "+-" on the root and the inverse would not turn out to be a function.)
root%283%2C+3x-3%29+=+y
4. We were able to solve for y. For each x, there is only one value for root%283%2C+3x-3%29. Our inverse is a function. So f(x) is one-to-one.