A function is a fundamental concept in mathematics. It's a special type of relation between inputs and outputs, where each input corresponds to exactly one output. Think of a function like a machine: you put in an input, and the machine gives you exactly one output.
To determine if something is a function, we use a simple test. A relation is a function if each input has exactly one output. If any input has multiple outputs, it's not a function. We can use the vertical line test: if a vertical line intersects a graph at more than one point, it's not a function.
Function notation uses f of x to represent a function. Here, f is the function name, x is the input variable, and f of x is the output. For example, if f of x equals 2x plus 3, and we input x equals 4, then f of 4 equals 2 times 4 plus 3, which equals 11. This notation helps us clearly identify inputs and their corresponding outputs.
There are many types of functions, each with unique characteristics. Linear functions have the form f of x equals mx plus b and create straight lines. Quadratic functions follow f of x equals ax squared plus bx plus c and form parabolas. Constant functions are simply f of x equals k, creating horizontal lines. Each type has distinct properties and graph shapes that help us identify and work with them.
To summarize what we've learned about functions: A function is a special relation where each input maps to exactly one output. We can identify functions using the vertical line test. Function notation like f of x helps us clearly express input-output relationships. Different types of functions create distinct patterns on graphs. Functions are fundamental mathematical tools used throughout science and engineering.