>>11868677>>11869028I'll try my best to explain but it might be below your level.
Like
>>11868721 said, an equation just states two things are equal.
If you give an equation an input, it will give you a single output. You can enter as many inputs as you want, but the equation will always just map one input to one output.
The function is what arrises when you ask, "what happens if we input *every* number?". It's the first abstraction in our story.
The function is what lets us see the forest for the trees. Since *every* conceivable input is defined by the function itself, we are then able to simply manipulate the symbolic function itself rather than having to manipulate every individual point. This allows us to derive and observe the properties of the functions in a much more simple, rigorous, and thorough manner.
The function allows us to see that, multiplying the function by some number 1/n squashes it by some factor n.
The function allows us to see that, if we add n to the input of the function before the value is computed, it shifts the graph up by n units.(f(x+n))
The function allows us to see that if we add n units to the computed value (f(x)+n), it will shift the graph left or right along the x-axis
Then, the concept of a function allows us to abstract even further. What does it mean to ask for the slope of an equation? An equation just gives us one single point. A single point has no slope, a point is unchanging. We can not define the slope of a simple equation.
The function gives us the tools to abstract the idea of a slope for any given input, since the function represents a smooth line rather than discrete points.
