A function is like a special machine. You put an input into the machine, and it gives you exactly one output. This is the key idea: for every single input value, a function produces one and only one output value.
Let's look at some 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 g of x equals x squared, input 2 gives output 4, and input 4 gives output 16. Notice the key rule: each input produces exactly one output.
To determine if something is a function, we use the vertical line test. If any vertical line crosses the graph more than once, it's not a function. A parabola is a function because each x-value has only one y-value. But a circle is not a function because some x-values have two y-values.
Function notation uses f of x to represent a function. The f is the function name, x is the input variable, and the expression tells us the rule. To evaluate f of 4, we substitute 4 for x: f of 4 equals 2 times 4 plus 3, which equals 8 plus 3, giving us 11.
To summarize what we've learned about functions: A function is a rule that assigns exactly one output to each input. We can test if something is a function using the vertical line test. Function notation helps us write and evaluate functions clearly. Functions are essential tools used throughout mathematics and science.