Function
A function is a mathematical relation where each input, often called the domain, is related to a single output, often called the range. Think of a function like a machine that takes an input and gives out exactly one output.
For instance, when you put a number into a function, you will always get a specific number in return. Consider a function that outputs the square of an input number. If you input the number 3, you get 9. If you input 4, you get 16. Every input corresponds to one unique output.
Input-Output Pairs
In the context of functions, input-output pairs are simply the pairs of numbers where the first number is the input and the second number is the output. In mathematical terms, these are represented as \(x,y\) pairs.
If you look at the exercise \(\{(0,-4),(2,-2),(4,0),(6,2)\}\), each pair indicates that when you input the first number into the function, you get the second number as output. For example, inputting 0 gives -4, inputting 2 gives -2, and so on. These pairs help us understand how inputs are mapped to outputs.
Unique Values
For a function to be termed one-to-one, each input must correspond to a unique output. This means no two different inputs give the same output.
In the example \(\{(0,-4),(2,-2),(4,0),(6,2)\}\), we can see that each input value \(0, 2, 4, 6\) corresponds to a unique output value \((-4, -2, 0, 2)\). No output value is repeated. This uniqueness is what guarantees that our function is one-to-one. Another way to put it is that if \(f(a) = f(b)\), then a must equal b.
Domain and Range
In functions, the domain is the set of all possible input values, while the range is the set of all possible output values.
Consider the example \(\{(0,-4),(2,-2),(4,0),(6,2)\}\). Here, the domain is \{0, 2, 4, 6\}, and the range is \{-4, -2, 0, 2\}. By listing out the domain and range, you can more easily check for uniqueness and map the relationship between inputs and outputs.
Understanding domain and range is crucial in understanding how functions behave and whether they are one-to-one.
