Math Problem Statement

Find the inverse of f(x) = (-4x - 5) / (4x - 3).

Solution

To find the inverse of the given function f(x)=4x54x3f(x) = \frac{-4x - 5}{4x - 3}, follow these steps:

Steps to find the inverse:

  1. Replace f(x)f(x) with yy: y=4x54x3y = \frac{-4x - 5}{4x - 3}

  2. Swap xx and yy: x=4y54y3x = \frac{-4y - 5}{4y - 3}

  3. Solve for yy: Multiply both sides by 4y34y - 3 to eliminate the denominator: x(4y3)=4y5x(4y - 3) = -4y - 5

    Distribute xx: 4xy3x=4y54xy - 3x = -4y - 5

  4. Group all terms involving yy on one side: 4xy+4y=3x54xy + 4y = 3x - 5

  5. Factor out yy: y(4x+4)=3x5y(4x + 4) = 3x - 5

  6. Solve for yy: y=3x54x+4y = \frac{3x - 5}{4x + 4}

Thus, the inverse function is: f1(x)=3x54x+4f^{-1}(x) = \frac{3x - 5}{4x + 4}

Let me know if you'd like more details on any step, and feel free to ask questions!

Related Questions:

  1. How do you verify if two functions are inverses of each other?
  2. Can every function have an inverse? What are the conditions?
  3. How does the behavior of a function change when you consider its inverse?
  4. What does it mean graphically when a function and its inverse are plotted?
  5. What role do one-to-one functions play in finding inverses?

Tip:

When solving for the inverse, always ensure the function is one-to-one (passes the horizontal line test) to confirm that an inverse exists.

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Math Problem Analysis

Mathematical Concepts

Functions
Inverse Functions

Formulas

Inverse Function Calculation: y = f(x), x = f(y)

Theorems

Function Inversion

Suitable Grade Level

College