A function is a mathematical relationship that assigns exactly one output value to each input value. Think of a function as a machine that takes inputs, processes them according to a specific rule, and produces outputs. For example, the function f of x equals x squared takes any number as input and outputs that number multiplied by itself. When we input 1, we get 1. When we input 2, we get 4. Each input has exactly one corresponding output.
Functions can be represented in multiple ways. The most common representations include equations like f of x equals x squared plus two x plus one, tables that list input-output pairs, graphs that provide a visual representation, and verbal descriptions. The key property of any function is that each input value must correspond to exactly one output value. This is often called the vertical line test for graphs - a vertical line should intersect the graph of a function at most once. As we move our input value along the x-axis, we can see how the function produces exactly one output value for each input.
Let's distinguish between functions and non-functions. Examples of functions include linear equations like f of x equals 2x plus 3, quadratic functions like g of x equals x squared, and trigonometric functions like h of x equals sine of x. In contrast, equations like x squared plus y squared equals 1, which represents a circle, are not functions because they fail the vertical line test. The vertical line test states that if any vertical line intersects a graph at more than one point, then the graph does not represent a function. This is because a function must assign exactly one output to each input. Looking at our examples, the parabola passes the vertical line test because each vertical line intersects it exactly once. The circle fails the test because a vertical line can intersect it at two points, meaning one input value would have two different outputs.
Two important concepts in functions are domain and range. The domain is the set of all possible input values for which the function is defined. The range is the set of all possible output values that the function can produce. Let's look at the function f of x equals the square root of x as an example. For this function, the domain is restricted to non-negative numbers, because we cannot take the square root of a negative number. So the domain is x greater than or equal to zero. The range is also restricted to non-negative numbers, because the square root operation always produces non-negative results. So the range is y greater than or equal to zero. Understanding domain and range helps us identify where functions are valid and what outputs they can produce.
To summarize what we've learned about functions: A function is a mathematical relationship that assigns exactly one output value to each input value. Functions can be represented in multiple ways, including equations, tables, graphs, and verbal descriptions. The vertical line test helps us determine whether a graph represents a function - if any vertical line intersects the graph more than once, it's not a function. Every function has a domain, which is the set of valid input values, and a range, which is the set of possible output values. Functions are fundamental concepts in mathematics and are used extensively in science, engineering, economics, and many other fields to model relationships between quantities.