A function is a relationship or rule that assigns exactly one output value to each input value. Think of a function as a machine that takes an input, processes it according to a specific rule, and produces exactly one output. For example, the function f of x equals x squared takes an input like 2, applies the squaring rule, and outputs 4. Every time you input 2, you'll always get 4 as the output. This is a key property of functions: the same input always gives the same output.
Functions can be represented in multiple ways. The most common representations include equations like y equals 2x plus 3, tables of values showing input-output pairs, graphs plotting the relationship, and mappings showing connections between inputs and outputs. A key characteristic of a function is that it passes the vertical line test: any vertical line drawn through the graph will intersect it at most once. This confirms that each input has exactly one output. For example, in the graph of f(x) equals x squared, the vertical line at x equals 1 intersects the graph at exactly one point, where y equals 1.
Every function has a domain and a 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. Since we can't take the square root of negative numbers, the domain is restricted to x greater than or equal to zero. And because the square root always gives non-negative results, the range is also y greater than or equal to zero. On the graph, the domain is represented by the green line along the positive x-axis, while the range is shown by the red line along the positive y-axis. The red shaded area represents values not in the domain of this function.
Functions come in many different types, each with unique properties. Common types include linear functions of the form f(x) equals mx plus b, which graph as straight lines. Quadratic functions like f(x) equals ax squared plus bx plus c form parabolas. Exponential functions like f(x) equals a to the power of x grow or decay rapidly. Other important types include logarithmic and trigonometric functions. Functions can also have special properties. A one-to-one function has each output coming from exactly one input, meaning the function is reversible. An onto function uses every possible output value in its range. These properties are important in advanced mathematics and real-world applications.
To summarize what we've learned about functions: A function is a rule that assigns exactly one output value to each input value. Functions can be represented in multiple ways, including equations, tables, graphs, and mappings. 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 come in many types, such as linear, quadratic, exponential, logarithmic, and trigonometric, each with unique properties and applications. Functions are fundamental tools used throughout mathematics, science, engineering, and many other fields to model relationships between quantities.