Writing an Inverse Function: Formulation Techniques

How to write an inverse function?

To write the inverse of a function, you follow a series of steps that involve algebraic manipulations and solving for the original variable. Here are the general techniques to formulate the inverse function:

Steps for Writing an Inverse Function:

1. Start with the Original Function:

• Begin with the original function, $f(x)$.

2. Replace $f(x)$ with $y$:

• Replace $f(x)$ with $y$, turning the equation into $y = f(x)$.

3. Swap $x$ and $y$:

• Swap the roles of $x$ and $y$, turning the equation into $x = f^{-1}(y)$.

4. Solve for $y$:

• Solve the new equation for $y$. The result is the expression for the inverse function, $f^{-1}(x)$.

Example:

Let $f(x) = 2x + 3$. Follow these steps:

1. Start with the Original Function:

   • $y = 2x + 3$

2. Swap $x$ and $y$:

   • $x = 2y + 3$

3. Solve for $y$:

   • $2y = x - 3$
   • $y = \frac{1}{2}(x - 3)$

So, the inverse function of $f(x) = 2x + 3$ is $f^{-1}(x) = \frac{1}{2}(x - 3)$.

Additional Tips:

• Verify the Inverse:

  • To ensure the correctness of your inverse function, check that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all $x$ in the domain.

• Graphical Interpretation:

  • On a coordinate plane, the graph of a function and its inverse are symmetric with respect to the line $y = x$.

• Domain and Range:

  • Be mindful of the domain and range when writing the inverse. In some cases, you may need to restrict the domain for the inverse to exist.

• One-to-One Functions:

  • For a function to have an inverse, it must be one-to-one (injective), meaning that each distinct input corresponds to a unique output.

By following these steps, you can write the inverse function of a given original function. Remember to verify your solution and consider any domain restrictions that may apply.

Procedures for writing an inverse function?