What are the domain and range of f – 1?

Inverse Functions: What’s In, What’s Out, and Why It Matters

Functions are the workhorses of mathematics, taking inputs and spitting out outputs. But what happens when you want to go the other way? That’s where inverse functions come in – they’re like the “undo” button for your math. Understanding how these inverses work, especially their domains and ranges, is super important. Trust me, it’ll save you headaches down the road. So, let’s dive in!

First Things First: Domain and Range – The Basics

Before we get to the fun part (inverses!), let’s quickly recap domain and range. Think of the domain as all the possible ingredients you can throw into your function machine – the x values that give you a valid result . The range, then, is everything that comes out of the machine – all the possible y values you can get .

Take f(x) = √x, for instance. You can only take the square root of zero or positive numbers, right? So, the domain is all non-negative numbers. And since the square root is always zero or positive, the range is also non-negative numbers . Simple enough!

Inverse Functions: Turning Things Around

An inverse function, usually written as f⁻¹(x), is all about reversing the process . It takes the output of the original function and gives you back the original input . Basically, if f(a) = b, then f⁻¹(b) = a .

Now, here’s a catch: not every function has a perfect inverse. To have an inverse, a function needs to be one-to-one. This means each input has to lead to a unique output – no duplicates! A handy way to check this is the horizontal line test: if any horizontal line crosses the graph more than once, it’s not one-to-one . Sometimes, if a function isn’t one-to-one to begin with, you can chop off part of it to make it one-to-one, and then find an inverse. More on that later.

The Big Secret: Domain and Range Swap!

Okay, this is the key takeaway:

• The domain of f⁻¹(x) is the range of f(x).
• The range of f⁻¹(x) is the domain of f(x).

Yep, that’s it! They just switch places . It makes sense when you think about it: the inverse function is just “undoing” what the original function did, so the inputs and outputs swap roles .

Finding the Domain and Range of an Inverse: A Step-by-Step Guide

So, how do you actually find the domain and range of an inverse function? Easy peasy:

Let’s do an example:

Say we have f(x) = x³.

That means:

• Domain of f⁻¹(x): All real numbers.
• Range of f⁻¹(x): All real numbers.

In this case, the inverse is f⁻¹(x) = ∛x, which confirms what we found.

Another Example, Just to Be Sure:

Let’s try f(x) = √(x – 1).

Therefore:

• Domain of f⁻¹(x): x ≥ 0.
• Range of f⁻¹(x): y ≥ 1.

To double-check, let’s find the inverse. If y = √(x – 1), then switching x and y gives us x = √(y – 1). Squaring both sides, we get x² = y – 1, so y = x² + 1. But remember that restriction! The domain of the inverse is x ≥ 0, so f⁻¹(x) = x² + 1 only when x ≥ 0. This confirms that the range is y ≥ 1.

One-to-One or Not One-to-One: That Is the Question

Like I mentioned earlier, only one-to-one functions have inverses over their entire domain. If a function isn’t one-to-one, we can sometimes “cut it down” to a piece that is one-to-one. A classic example is f(x) = x². It’s not one-to-one because both 2 and -2 give you 4. But if we only look at x values that are zero or more (x ≥ 0), then it’s one-to-one, and its inverse is f⁻¹(x) = √x.

Wrapping It Up

Understanding the domain and range of inverse functions might seem a bit abstract, but it’s a fundamental concept in math. The key is remembering that the domain and range simply swap places between a function and its inverse. Keep an eye out for functions that aren’t one-to-one and might need a little “trimming” to find their inverse. Once you get the hang of it, you’ll be able to navigate the world of functions and their inverses with confidence!