IME explanations of the Fourier transform focus far too much on the specific mechanics of it without motivating the reason. I find people often don't even realize the time space and frequency space functions are the same function! I dive in like so
"There are many different ways to write down the number 5. Tally marks, the sigil 5, 4+1, 2.5 times 2, 10/2. Each of them is more or less useful in different circumstances. The same is true of functions! There are many different ways to write the same function. The mechanics of taking a function in one form and writing it in a new form is called a transformation."
Above needs to be said a hundred times. Then you talk about why a frequency space function can be useful. Then you talk about how to transform a time space function to the frequency space representation.
"There are many different ways to write down the number 5. Tally marks, the sigil 5, 4+1, 2.5 times 2, 10/2. Each of them is more or less useful in different circumstances. The same is true of functions! There are many different ways to write the same function. The mechanics of taking a function in one form and writing it in a new form is called a transformation."
Above needs to be said a hundred times. Then you talk about why a frequency space function can be useful. Then you talk about how to transform a time space function to the frequency space representation.