A function is a special relationship between inputs and outputs. The key rule is that each input has exactly one output. Think of it like a machine: you put in an input x, and the function machine gives you exactly one output f of x.
Let's look at specific examples. For the function f of x equals 2x plus 1, when we input 3, we get output 7. When we input 5, we get output 11. For the function g of x equals x squared, input 2 gives output 4, and input negative 3 gives output 9. Notice how each input always produces exactly one output.
To determine if a relationship is a function, we use the vertical line test. If any vertical line crosses the graph at more than one point, then it's not a function. In the first example, each input x value has exactly one output y value, so it's a function. In the second example, the input x equals 1 has two different outputs, so it's not a function.
Function notation is a standard way to write functions. In f of x equals 2x plus 1, f is the function name, x is the input variable, and f of x represents the output value. We read this as 'f of x equals 2x plus 1'. When x equals 2, we calculate f of 2 equals 2 times 2 plus 1, which equals 5.
To summarize what we've learned about functions: A function assigns exactly one output to each input. We can use the vertical line test to determine if a relationship is a function. Function notation like f of x helps us express input-output relationships clearly. Functions are fundamental tools that appear throughout mathematics and real-world applications.