The way a function can be looked at as a machine is due to the fact that machines require supplies to be placed in it to make sure goods are produced out of it. In order to have good products formed in the machine needs to follow a format and have the proper supplies given to it. That is the same way functions work. One starts off with a formula to follow, which consist of at least one variable in the formula. This is known as the input of this machine and the output is the solution to the formula. One must find a number that works with the formula in order to have it listed as one of its domains. A domain is defined as number sets of the input that makes the output real.
The range of a function is the difference between the lowest and highest values. The way the range of a function relates to a functions formula is when an individual wants to find out what is the highest and lowest input that can make the output real.
An example will be from the function machines app on Geogebra. To make all the functions have the same output one used the lowest range each formula had to have the same output. This is shown in the image below.
I could not find a number that cannot be the output of all four functions. For formula F(x), H(x) and P(x) I found numbers -3,3 and 5, but ould not find a number for G(x).
Has there ever been a time where an input/output function was useful in a real life scenario? A time for me was when I had to figure out the amount of exercise and calorie count in a healthy diet to fit a dress for an event. (silly I know,) Now can anyone else think of one?