A function is a fundamental concept in mathematics. It represents a special relationship between inputs and outputs, where each input value corresponds to exactly one output value. Think of it like a machine that takes an input, applies a specific rule, and produces a unique output.
Functions have several key components. The domain is the set of all possible input values, while the range is the set of all possible output values. We use function notation like f of x equals 2x plus 1 to represent functions. For example, when x equals 3, f of 3 equals 2 times 3 plus 1, which equals 7.
Functions can be represented in multiple ways. First, we have equation form, like f of x equals x squared minus 2x plus 1. Second, we can use a table of values showing specific input-output pairs. Third, we have the graph form, which provides a visual representation of the relationship between inputs and outputs.
Functions can be combined through various operations. We can add functions, multiply them, or compose them. Function composition means applying one function to the result of another. For example, if f of x equals x plus 1 and g of x equals 2x, then f composed with g of x equals f of 2x, which equals 2x plus 1.
To summarize what we have learned about functions: Functions are mathematical relationships that map each input to exactly one output. They have key components like domain and range, can be represented in multiple ways, and can be combined through various operations. Functions are fundamental tools used throughout mathematics and science.