How To Find The Inverse of a Function: Here’s how!

How To Find The Inverse of a Function
High school student

If you are a student who has been struggling with finding the inverse of a function, you are not alone. Many students find it difficult to find the inverse of a function. The truth is that finding inverses is an essential component of calculus, and mastering it can help you solve complex mathematical problems with ease.

So, how to find the inverse of a function? First, represent the function as y = f(x). Secondly, swap the positions of x and y in the function, resulting in x = f(y). Then, solve for y to get the inverse function. An essential step in solving for y involves solving for it while treating x as the subject of the equation.

Read on to explore what the inverse of a function is, why it is important, and how you can find it step by step. If, like most students you wonder whether ChatGPT can help you do Calculus or not, the answer is yes, to find how, I encourage you to read this article.

What Is the Inverse of a Function?

An inverse function is a function that undoes another function. In simpler terms, if you have a function f(x) and apply g(x) to it, you get back x. In other words, g(x) is the inverse of f(x). If f(x) produces y, then putting y into the inverse of f creates the output x. A function f with an inverse is called invertible, and the inverse is denoted as f−1(x) (Source: Simon Fraser University)

The arrow diagram below illustrates inverses of function.

Why Find the Inverse of a Function?

Finding the inverse of a function is essential because it can help in solving various mathematical problems. It is used in differential calculus, trigonometry, linear algebra, and other mathematical branches.

Moreover, the inverse of a function can help in solving equations, finding the domain and range of a function, and exploring the symmetry of graphs. If you want to learn more about the inverse of a function, I encourage you to check out this excellent article from Khan Academy.

Steps to Find the Inverse of a Function

The process of finding the inverse of a function can be broken down into several simple steps. To start with, suppose we have a function f(x). The inverse of the function is a new function that reverses the input/output relationship of the original function. In other words, if we have an input, say y, the inverse function will give us the output, x.

Step 1: Finding the Inverse of a Function is to Write it In The Form of y = f(x)

The first step to finding the inverse of a function is to write it in the form of y = f(x).

For example, let’s take the function f(x) = 2x – 3. To find its inverse, we need to rewrite it as y = 2x – 3.

Step 2: Switch x and y

The second step is to switch x and y. In our example, we get x = 2y – 3. This second equation now represents the inverse function.

Step 3: Solving for y by Isolating it on One Side of The Equation

Next, we solve for y by isolating it on one side of the equation. In our example, we start by adding 3 to both sides of the equation, which gives us x + 3 = 2y.

We then divide both sides by 2 to obtain y on one side, giving us the inverse function y = (x + 3)/2.

There, f-1(x) = (x + 3)/2

It is essential to verify the inverse of the function by using the composition of functions by substituting the inverse function into the original function and then vice versa.

Finding An Inverse Function Formula

Given a formula for f(x), we can find a formula for f −1 (x), using the following equivalence:

x = f−1(y) if and only if y = f(x)

Generally, we can find a formula for f−1 using the following method:

  • In the equation y = f(x), if possible, we solve for x in terms of y to get a formula x = f-1(y).
  • Then, we switch the roles of x and y to obtain a formula for f-1 of the form y = f-1 (x).
How To Find The Inverse of a Function
Finding the inverse of a function

Tips for Finding the Inverse of a Function

While finding the inverse of a function may seem overwhelming, there are a few tips to make the process easier. One tip is to choose functions that are one-to-one functions.

1- Choosing Functions That are One-to-One Functions

  • A one-to-one function is a function where no two x’s result in the same y. A one-to-one function is a particular function that maps every element of the range to precisely one part of its domain, meaning that the outputs never repeat. 

2- Use The Horizontal Line Test

Another way to check if a function has an inverse is to use the horizontal line test. A function is said to have an inverse if and only if every horizontal line intersects at most once with its curve.

If there is a horizontal line that intersects the curve of a function more than once, then the function does not have an inverse. In other words, a graph passes the Horizontal line test if every horizontal line cuts the graph at most once (Source: University of Notre Dame)

Not all Functions Have an Inverse

Remember that not all functions have an inverse. For a function to have an inverse,

  • It must be a one-to-one function, which means that it maps distinct inputs to distinct outputs.
  • If a function fails the horizontal line test, then it is not one-to-one and does not have an inverse.

Practice, Practice, Practice

Like any mathematical concept, the more you practice, the better you’ll understand it. Practice finding the inverse of functions until you feel confident in your skills. I encourage you to practice finding the inverse of functions to challenge yourself and sharpen your skills.

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Wrapping Up

The inverse function is a fundamental concept in mathematics that is used in various applications. I believe that by following the steps outlined in this blog, finding the inverse of a function should be a little less intimidating.

Remembering to use the horizontal line test to confirm whether a function has an inverse or not.

Altiné

I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.

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