Operations on functions such as f+g, f(g(x)), f(x+e) will be important as we cover derivatives, so we cover it now. We define functions 3 ways: an input-output table, a formula defining an input-output rule, and a graph. Sometimes it might be convenient to call the input-output table as a "function box".
In exercises, x often has dual contexts and this is a major cause of confusion. Here are 2 examples:
- y = f(x), f'(x) = lim f(x+e) - f(x) / e
In the first context, x is the input of a function, but in the second context, x is only a partial input.
- y = f(x), z = g(x), f(g(x))
In the first context, x is the input of f, but in the second context, x is not the input of f; rather, the input is g.
We recommend avoiding confusion by these dual context, by using separate variables:
- y = f(u), f'(x) = lim f(x+e) - f(x) / e
- y = f(u), z = g(v), f(g(x))
In other words,
favor this: g f v --> 2v (=u) --> cos 2v or this: f(u) = cos u, g(v) = 2v, f(g(v)) = cos 2v instead of: f(x) = 2x, g(x) = cos x, f(g(x)) = cos 2x
A similar confusion exists when we name functions in a formula such as the chain rule, and use the same name in a slightly different context of an exercise. For instance, f'(g(x))*g'(x) and the exercise 1 / cos x, where g=1/x and f = cos x.
Even a formula as symmetric as the product rule can be confusing: (fg)' = f'g + g'f, and the exercise would read h(x) = cos x, k(x) = sin x. The students may not know how to use the formula because he/she expects f & g to appear in the exercise.
To avoid such confusion, we should we state formula with a worded description and avoid names like f,g,x,u,v... Once worded, then can supplement the description with formulas w/ named functions & variables. Therefore the Chain Rule should be stated as:
- speed of outer function * speed of inner function.
Similarly, we state the product rule as:
- sum of conjugates
- "speed of one function * the other + its conjugate
- "speed of one function * the other + speed of the other function * the one"
- f'g + g'f
Another pitfall from dual notation is that of the parentheses' role as multiplication, and input of a function.
In Linear Systems, we introduced shapes as a notation for the dependent variable y. We continue that notation here for the function f(x) to simplify composition of functions.
Notation: +-------+ f(x) is written as a rectangle | ..x.. | +-------+ and its input written as ".." in the shape's interior. Thus following composition f(u), u(x)=cos2x, f(u(x)) = f(cos 2x) is more concisely drawn as +-----------+ |.. cos2x ..| +-----------+ ".." means that cos2x is the INPUT of the rectangular function +------------+ Not to be confused with | cos2x | +------------+ no ".." means the rectangular function is the SAME as its interior
The more symbols we eliminate, such as the intermediary u(x), the better.
Shapes even simplify calculus. Say we use shapes to represent a function, and a typographical coding, such as a dotted border shape for the derivative of a function and dashed border for its antiderivative, the chain rule is easier to read.
standard notation: d -- f(g(x)) = f'(g(x)) g'(x) dx shape notation: +------+ :::::::: d | /\ | : /\ : : -- | \/ | = : \/ : : : dx +------+ :::::::: :
It is easier for a child to distinguish a solid triangle from a dashed triangle than to distinguish g'(x) from g(x).
In Calculus, we will build new functions from smaller functions f & g: f+g, f*g, f/g, f(g(x)). In proving the derivative identities, it is vital that students understand this notation at an instinctive level, for example, they must know that f+g is a new function h and that
h(x+e) = (f+g)(x+e) = f(x+e) + g(x+e). x+e is the input of f+g, not multiply
In such cases, shape notation help tremendously. Make sure they can complete the following table in a blink of an eye:
|input||f||g||h = f+g||h = f*g||h = f/g||h = fog (function composition)|
Functions are best defined as input-output mappings. But other definitions as graphs and rules are helpful.