Inverses
What is an inverse? Why is it important?
If inverse is a new word to you, that's ok! I want to introduce this before diving into the Nspire activity because this term will come up again there. It's best that you have some kind of intuition about this idea before continuing.
We will start with a simple example of inverses:
Think of addition:
We begin with the number 2 and add 3: 2+3 = 5
Now, what if we start with the number 5 and want to go back to 2? What do we have to do?
5-3 = 2. Thus, the inverse of addition is subtraction. This helps us go back to the original number that we started with.
Similarly, multiplication is the inverse of division. We can think of this as an operation that "undoes" another operation.
If we have 5*2 = 10, then we use multiplication.
If we start with 10, how do we go back to 5? We divide by 2! Thus, multiplication and division are inverse operations.
Now, let's think about inverses with functions. When we think of inverses, we should think about a function "undoing" the other. This can also be rephrased as a function f(x) and a function g(x) such that their composition equals x. Huh? Here's a visual to understand this better.
If inverse is a new word to you, that's ok! I want to introduce this before diving into the Nspire activity because this term will come up again there. It's best that you have some kind of intuition about this idea before continuing.
We will start with a simple example of inverses:
Think of addition:
We begin with the number 2 and add 3: 2+3 = 5
Now, what if we start with the number 5 and want to go back to 2? What do we have to do?
5-3 = 2. Thus, the inverse of addition is subtraction. This helps us go back to the original number that we started with.
Similarly, multiplication is the inverse of division. We can think of this as an operation that "undoes" another operation.
If we have 5*2 = 10, then we use multiplication.
If we start with 10, how do we go back to 5? We divide by 2! Thus, multiplication and division are inverse operations.
Now, let's think about inverses with functions. When we think of inverses, we should think about a function "undoing" the other. This can also be rephrased as a function f(x) and a function g(x) such that their composition equals x. Huh? Here's a visual to understand this better.
This means that f(g(x)) = g(f(x)) = x.
Let's take a look at this closer.
Here is a desmos activity that will help demonstrate inverses of functions visually (Thielen).
We see that our first function is f(x) = x^2. This can be rewritten as y = x^2. Let's swap the x and y variables, so now we have x = y^2.
Now, solve for y. What do you get? You can check your answer with the second function in our list. If we graph both of them, how do they compare to one another?
Continue through this activity until you have looked at all four pairs of functions.
Hopefully, now you are more comfortable with what inverse functions are and are ready to move on to using the Nspire calculator! If so, click here.
Let's take a look at this closer.
Here is a desmos activity that will help demonstrate inverses of functions visually (Thielen).
We see that our first function is f(x) = x^2. This can be rewritten as y = x^2. Let's swap the x and y variables, so now we have x = y^2.
Now, solve for y. What do you get? You can check your answer with the second function in our list. If we graph both of them, how do they compare to one another?
Continue through this activity until you have looked at all four pairs of functions.
Hopefully, now you are more comfortable with what inverse functions are and are ready to move on to using the Nspire calculator! If so, click here.