 A function is an operation or a series of operations that transforms a set of numbers into a different set of numbers. An input is put into the function, the operations are performed and a different number is outputted. So if we input 3 into our function, multiply by 5, subtract 2, then 13 is outputted. 3 has transformed into 13. Mathematically, we'd write this function as f of x equals 5x minus 2, where x is our input and what f of x equals is our function. f of x is the mathematical notation. Think of it in a similar way to how you'd think of y. We say a function of x equals 5x minus 2. If we wanted to calculate the value of f of 3, we just need to replace the x in the function with 3, which we already knew from our input output machine earlier. So what's the value of f of 4 for this function? Did you get f of 4 equals 8? What about this question? It's slightly different. This time we're not being asked to substitute in a value of x, but instead are being told the function equals 35, so the output is 35. It wants us to find x to find the input. Pause the video, solve for x and click play when you're ready. Did you get x equals negative 5? So an input of negative 5 will give an output of 35. We usually use the letter f, but you can actually use any letter. You'll see g of x, h of x, they all just mean a function of x equals. There are two special rules which determine whether something is a function or not. A function must work for every input value and every input value can only give one output value. This is not a function because when we input negative 4 and negative 9, we don't get an output. This would be a function because every input of x has an output. This is also a function because inputs can share outputs, but this would not be a function because an input of 1 can't have multiple outputs. So the final piece of the functions puzzle that we need to address in this introduction video are domains and ranges. We will come across them a lot in functions. The domain just means the inputs. The domain can be a range of values, so here it's 1, 4, 6 and 7. The range are the output values from the function. So what comes out when we put our domain values in? Let's send our introduction to functions with these two questions. Pause the video, work out the answers and click play when you're ready to check. How did you get on? Did you get the function of h equals negative 6 when the input is negative 4? And x equals 3 when function f and function g are equal? So that's our introduction to functions. We've learnt what they are, we've looked at the notation and we've answered a few function questions. We've discovered that the domain means the inputs and the range means the outputs. With a little side note, functions can also be written like this. It just means the same thing, so don't stress if you see that notation. In future videos we'll look at composite functions, inverse functions and the graphs of functions.