Understanding Inverse Functions

What Are Inverse Functions?

An inverse function is like reversing a process. Imagine a function as a machine that transforms an input $(x)$ into an output $(y)$. The inverse function lets you reverse this process, turning the output back into the input.

For example:

• If $f(x) = x + 3$, its inverse function, written as $f^{-1}(x)$, undoes the process: $f^{-1}(x) = x + 3$

How Are Inverse Functions Written?

We write the inverse of a function $f(x)$ as $f^{-1}(x)$ (read as "f inverse of x"). It is important to note that $f^{-1}(x)$ is not the same as $\frac{1}{f(x)}$.

 

Characteristics of Inverse Functions

• Reversal of Roles: In the inverse function, the domain and range of the original function switch places.
• Graphical Symmetry: The graph of an inverse function is a reflection of the original function's graph across the line $y = x$.
• Not Always Exist: For a function to have an inverse, it must be bijective (both injective and surjective), meaning it's both one-to-one and onto.

 

Steps to Find an Inverse Function

1. Write the function as $y = f(x)$: Replace $f(x)$ with $y$
2. Swap $x$ and $y$: Interchange $x$ and $y$ in the equation
3. Solve for $y$: Rearrange the equation to make $y$ the subject. This new equation is $f^{-1}(x)$, the inverse function
4. Rewrite using inverse notation: Replace $y$ with $f^{-1}(x)$.

 

Examples

Example 1: Find the Inverse of $f(x) = 2x + 5$

Solution:

• Rewrite $f(x)$ as : $$y = 2x + 5$$
• Swap $x$ and $y$: $$x = 2y + 5$$
• Solve for $y$: $$\text{ Subtract 5 from both sides: } \ x - 5 = 2y \\  \text{Divide by 2: } \ y = \frac{x - 5}{2}$$
• Rewrite using inverse notation: $$f^{-1}(x) = \frac{x - 5}{2}$$
  
  

 

 

 

 

 

 

Example 2: Find the Inverse of $f(x) = \frac{x + 3}{2}$

Solution:

• Rewrite $f(x)$ as : $$y = \frac{x + 3}{2}$$
• Swap $x$ and $y$: $$x = \frac{y + 3}{2}$$
• Solve for $y$: $$\text{Multiply both sides by 2: } \ 2x = y + 3 \\ \text{Subtract 3 from both sides: } \  y = 2x - 3$$
• Rewrite using inverse notation: $$f^{-1}(x) = 2x - 3$$