A function is a relation where each input has exactly one output.The range of a relation is the collection of the second entries of each ordered pair. The domain of a relation is the collection of the first entries of each ordered pair.A relation can also be visualized using a table or mapping diagram.A relation is a collection of ordered pairs.Order is important! These numbers cannot be swapped and still have the same meaning. The first number is often represented with an \(x\) and the second with a \(y\). An ordered pair relates one number to another number.To evaluate a function \(f\) that uses an equation for a rule, we take the input and swap it out for \(x\) in the rule. For example, \ For this function, the rule is that we take the input number that \(x\) represents, and then multiply it by 2. This allows us to evaluate a function for a much larger collection of numbers than in our previous examples. This is how a function pairs numbers in its domain with numbers in its range.įor the rest of this course, the rule that a function follows will be given by an equation using a variable. Something goes in (input), then something comes out (output). The machine receives input, processes it according to some rule, then outputs a result. It is helpful to think of a function as a machine. In \(a.\), for example, this notation is read as "\(f\) of 1 is equal to 3", meaning that when we evaluate \(f\) with the input 1, the output is 3. Note: in this setting, the parentheses do not represent multiplication we are not multiplying the function \(f\) by the input. In other words, if we want to know what a function does to a given input, we indicate the function with \(f\), place the \(input\) in parentheses, and set it equal to the corresponding output. If \(f\) is a function, function notation looks like \ Or if we want to use \(x\)'s and \(y\)'s, it is common to say \ In the sections that follow, we will explore other ways of describing a function, for example, through the use of equations and simple word descriptions.īefore we can talk about writing the rule for a function using an equation, we must define a new notation that we call function notation. For example, we can use sets of ordered pairs, graphs, and mapping diagrams to describe the function. One way to think about a function is as being a rule that pairs one input with one output.
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