# Invertible Function Worksheets

Inverse functions are the polar opposite or reverse of one another. Meaning that they can be used to cancel one another out, when using complex math operations. If a function has an inverse and each individual input has a unique output we called it invertible. We can test this property by using the vertical line test on a graph of it. If we hit it in more than one section, it is non-invertible. These worksheets and lessons look at special functions that are unique in that each input has a unique output. We will learn how to compose and decompose functions that exhibit this specific property.

### Aligned Standard: HSF-BF.B.4d

- Inverse Functions Step-by-step Lesson-See if you can find the inverse of the given function.
- Guided Lesson - These problems have several steps for you to run through to get them prepared for the final step.
- Guided Lesson Explanation - Swapping out the x and y variables help here a lot.
- Practice Worksheet - You will find all four standard formats of functions to work with here.
- Matching Worksheet - Don't forget about the notation used for an inverse of a function.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

We start with very deliberate values for f(x).

- Homework 1 - Swap the x and y variables to create the inverse relation will be the set of ordered pairs.
- Homework 2 - Divide both sides by 5: x+3 = 5y and swap sides.
- Homework 3 - Find the inverse of the function: f(x)=x+3/x

### Practice Worksheets

We introduce multiple operations in these problems.

- Practice 1 - Determine the inverse of this function. Is the inverse also a function?
- Practice 2 - What is the inverse of the function f(x) = 5x-7
- Practice 3 - These are more about noticing a pattern.

### Math Skill Quizzes

These function table questions might take a little extra time.

- Quiz 1 - Since function ƒ was a one-to –one function (no two points share the common value), the inverse relation will be a function.
- Quiz 2 - You will need to do a bit of algebra here.
- Quiz 3 - I wish you put it all together here.

### What are Invertible Functions?

Just about everything has an opposite, like day and night, hot and cold, shiny and dull, functions also have opposites or inverses. If a function does have an inverse, we can refer to that as an invertible function. It is important to note that not every function has an inverse. In the most general sense, functions that reverse each other are known as inverses. A function is known as invertible only if each of your inputs has a unique output. To clarify, each output is paired with exactly one input so that when you reverse the input, it will still be a function. In order for a function to exhibit these characteristics, it must exhibit both the one-to-one and onto properties.

Let's take a look at what this looks like with math. If we were to consider the function; f(y) = 4y + 6. Then, the inverse of the function will be f^{-1}(y)= y^{-6/4}. If we were to graph these functions it would result in a wavy line and mirror image of it across the origin on the other side of the graph. We can use form of math to help use compose a function, but also decompose them to. This can also be used to help us isolate sections of interest within the math found behind it.