Math Problem Statement

How do I find the inverse of this

How do I find the inverse of this?

Solution

To find the inverse of the function in the given image, let’s break down the process step by step. The function provided is:

f(x)=3x42xf(x) = \frac{-3x - 4}{2 - x}

We will determine the inverse f1(x)f^{-1}(x) as follows:


Steps to Find the Inverse:

  1. Replace f(x)f(x) with yy: y=3x42xy = \frac{-3x - 4}{2 - x}

  2. Swap xx and yy: This step is crucial because the inverse function reflects xx and yy in the equation: x=3y42yx = \frac{-3y - 4}{2 - y}

  3. Solve for yy: Rearrange the equation to isolate yy:

    • Multiply both sides by (2y)(2 - y) to remove the denominator: x(2y)=3y4x(2 - y) = -3y - 4
    • Expand: 2xxy=3y42x - xy = -3y - 4
    • Move all terms involving yy to one side: 2x+4=xy3y2x + 4 = xy - 3y
    • Factor out yy on the right-hand side: 2x+4=y(x3)2x + 4 = y(x - 3)
    • Solve for yy: y=2x+4x3y = \frac{2x + 4}{x - 3}

Final Answer:

The inverse function is: f1(x)=2x+4x3f^{-1}(x) = \frac{2x + 4}{x - 3}


Let me know if you'd like me to verify the result or if you have any additional questions! Below are some related problems for practice.


Related Questions:

  1. How can you confirm that f(x)f(x) and f1(x)f^{-1}(x) are truly inverses of each other?
  2. What is the domain of f(x)=3x42xf(x) = \frac{-3x - 4}{2 - x}?
  3. What is the domain of f1(x)=2x+4x3f^{-1}(x) = \frac{2x + 4}{x - 3}?
  4. If f(x)=3x42xf(x) = \frac{-3x - 4}{2 - x}, what is f(1)f(1)?
  5. How does the graph of f(x)f(x) relate to the graph of its inverse?

Tip:

When solving for the inverse, always check for restrictions in the domain to avoid division by zero or other undefined operations.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Rational Equations
Algebra

Formulas

y = (ax + b) / (cx + d)
Inverse formula by swapping x and y: x = (ay + b) / (cy + d)

Theorems

Definition of an Inverse Function

Suitable Grade Level

Grades 10-12