Determine whether each function is a one-to-one function. If...

Determine whether each function is a one-to-one function. If it is one-to-one, list the inverse function by switching coordinates, or inputs and outputs. See Examples I and 3.
$$
f=\{(-1,-1),(1,1),(0,2),(2,0)\}
$$

Key Concepts

One-to-One Function

A one-to-one function, also known as an injective function, is one in which every element of the function’s domain is mapped to a unique element in the range. This means that no two distinct inputs produce the same output, ensuring that the function has a unique correspondence between its domain and range.

Inverse Function

An inverse function is a function that reverses the mapping of the original function, switching the roles of inputs and outputs. To find the inverse, one exchanges the coordinates of each ordered pair of the original function, provided the function is one-to-one, so that every output is uniquely mapped back to an input.

Transcript

00:02 Hello, today we're going to determine whether this function here is a one -to -one function.

00:06 If it is one -to -one, we will list the inverse function by switching the coordinates or inputs and outputs.

00:13 So here we have function notation, f equals, we have our set, our curly brackets.

00:19 Inside of the set, we have coordinate points, our x and our y values.

00:24 X is first our inputs, comma, y, our outputs.

00:29 For it to be a function, remember, each input can only have one unique output.

00:35 So if we look at negative one, it only has one output of negative one.

00:38 Great.

00:39 There's no other coordinate point here that has a negative one for x with a different output than this one here.

00:47 Now if we look at the input of one, that has an output of one, the input of zero, that has an output of two, the input of two output of zero.

00:54 Great.

00:56 This is a function.

00:58 Now for it to be a one -to -one function specifically, all the outputs need to map or have only one unique corresponding input, basically.

01:12 So i'm going to draw a mapping circle to show what's going on here.

01:16 Now i'm going to put all my y values on the left and my x values on the right.

01:20 Let me first show you what a non -example would be.

01:24 For example, if i had a 3 here and i had a y value of 3 and that was corresponding to two inputs, that would not be a one -to -one function.

01:37 It might be a function normally, but it wouldn't be a one -to -one function...

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