When is Multiplying or Composing Functions Useful?

[armandshamar]

I know this sounds like a strange, maybe impossibly naive question, but I have a B.S. in Math and have done some graduate work and I find myself teaching the chain rule and product rule to undergrads. I can get some intuition on why the rules work the way they do, but I find I can't think of any good simple concrete examples of why we would have functions for things and then multiply or compose those functions. What are some nice examples of this?

 

[BucketOfFish]

Suppose you have some machinery in which there are rotating components, like gears or cams. Maybe like a drivetrain in a car. To find the torque on a component, you want to multiply force by axial distance. But both of these change as your component rotates through its cycle, and so are dependent on the angle. So to find the torque at any time, you multiply one function of angle (force) by another function of angle (axial distance).

Now think about the engine in the car. The fuel efficiency of the car depends on the efficiency of the engine, and the engine efficiency depends on the temperature. So the fuel efficiency is a function of the engine efficiency, which is a function of the temperature.

These concepts are extremely general, and I can literally think of thousands of other examples. Whenever you multiply two values, ask yourself whether these values depend on any variables. If so, you are multiplying two functions together. Whenever you have one function, ask if the variables themselves depend on other variables. If so, you have a function of a function.

I hope this helped; ask if you still have questions.

 

[HallsofIvy]

The most common use of "composition" is to break complicated functions into simpler functions. For example, $\sqrt{1- x^2}$ can be thought of as the compostion of $f(x)= \sqrt{x}$ and $g(x)= 1- x^2$.

 

[Redbelly98]

The classical damped (decaying) harmonic oscillator in physics has solutions of the form
$$x(t) = e^{-\alpha t} \left( A \sin(\omega t) + B \cos(\omega t) \right)$$
If you want to get the velocity and acceleration, you'll need to use the product rule when you differentiate.