I’m having some trouble understanding functions. Mainly concerning how to know what the input and outputs are. Could you give me an explanation?

The best way to think about a function is to see it as a machine. Your input is the item you put into the machine, the function is what the machine does, and then when you get out of the other side is your output. For example think of a cupcake icing machine. Your input you put into the machine would be the cupcake, the function would then be the action of putting the icing on the cake, and your output would be the finished product of the iced cupcake. Now let’s try to expand this idea to an examinable question. Let’s say we have a function x^2 + 1, this means that our machine will take a number x which will be our input, square it and then add 1 to it. If our input is x=2 what would our output be? To do this we put our input into the machine, and the machine will carry out the function. 2^2 is 4 and then adding 1 to this will give us 5. Therefore we have our output which is 5. You could also be asked to work backwards. For example the question could give us the output and ask us to find the input. Using the same example, if our output is 10, we need to undo what the machine has done to find our input x. So instead of squaring and adding 1, we will take away one and then take the square root. Taking away 1 gives us 9 and then our square root is 3, and we have found our input x=3. To test if this answer is correct we can put our input back into the machine and see if we get the same output. So going back in we have 3^2 which is 9, add 1 which is 10. Therefore our input x is correct.