Math Problem Statement

f^(-1)(x)=(1)/(\sqrt(3-2x))

Solution

You have given the function f1(x)=132xf^{-1}(x) = \frac{1}{\sqrt{3 - 2x}}, which represents the inverse of some original function f(x)f(x). Now, if we want to find the original function f(x)f(x), we need to follow these steps to reverse the inverse process.

Steps to find f(x)f(x):

  1. Start with the inverse function: y=132xy = \frac{1}{\sqrt{3 - 2x}} where y=f1(x)y = f^{-1}(x).

  2. Swap xx and yy to reverse the function: x=132yx = \frac{1}{\sqrt{3 - 2y}}

  3. Solve for yy (which will now represent f(x)f(x)):

    Multiply both sides by 32y\sqrt{3 - 2y} to eliminate the denominator: x32y=1x \sqrt{3 - 2y} = 1

  4. Divide both sides by xx: 32y=1x\sqrt{3 - 2y} = \frac{1}{x}

  5. Square both sides to remove the square root: 32y=1x23 - 2y = \frac{1}{x^2}

  6. Isolate yy by subtracting 3 from both sides: 2y=1x23-2y = \frac{1}{x^2} - 3

  7. Finally, divide by 2-2 to solve for yy: y=31x22y = \frac{3 - \frac{1}{x^2}}{2}

Thus, the original function is: f(x)=31x22f(x) = \frac{3 - \frac{1}{x^2}}{2}

Would you like further details on any part of this process, or do you have any questions?

Here are some related questions for practice:

  1. What is the domain of the function f1(x)f^{-1}(x)?
  2. How would you verify that f(f1(x))=xf(f^{-1}(x)) = x?
  3. What are the restrictions on xx for the original function f(x)f(x)?
  4. How does the graph of f(x)f(x) compare to the graph of f1(x)f^{-1}(x)?
  5. Can f(x)f(x) or f1(x)f^{-1}(x) have any horizontal asymptotes?

Tip: When dealing with inverse functions, always check the domain and range carefully since they are swapped between the function and its inverse.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Algebra
Radical Equations

Formulas

y = 1/√(3 - 2x)
f(f^{-1}(x)) = x

Theorems

Properties of Inverse Functions
Square Root Properties

Suitable Grade Level

Grades 10-12