A function is a special mathematical relationship between inputs and outputs. The key rule is that each input must have exactly one output. Think of a function as a machine: you put in a number, and it gives you exactly one result back. For example, if we have the function f of x equals x squared, when we input 3, we always get 9 as the output.
Functions use a special notation to show the relationship clearly. We write f of x equals some expression. The letter f is the function name, x is the input variable, and the expression on the right shows the rule. For example, f of x equals 2x plus 1 means we multiply the input by 2 and add 1. We can use any letter for the function name like g or h.
Evaluating a function means finding the output for a specific input value. Let's use the function f of x equals x squared plus 1. To find f of 2, we replace x with 2. So f of 2 equals 2 squared plus 1, which equals 4 plus 1, which equals 5. On a graph, this means when x is 2, the y value is 5.
To determine if a relationship is a function, we use the one-to-one rule: each input must have exactly one output. On the left, we have a parabola which is a function because each x value gives only one y value. On the right, we have a circle which is not a function because some x values give two y values. We can use the vertical line test: if any vertical line crosses the graph more than once, it's not a function.
To summarize what we have learned about functions: A function is a special mathematical relationship where each input has exactly one output. We use notation like f of x to represent this relationship clearly. To evaluate a function, we substitute the input value and follow the mathematical operations. The vertical line test helps us determine if a graph represents a function. Functions are fundamental tools that appear throughout mathematics and have countless real-world applications.