Calculus: Project 10

The use of "unknown functions" is of fundamental importance in calculus and other branches of mathematics and science. We shall often see in the course that differential equations can be viewed as identities for unknown functions. One reason that students sometimes have difficulty understanding the meaning of derivatives or even the general rules for finding derivatives is that those things involve equations in unknown functions. The symbolic rules for differentiation and the increment approximation defining derivatives are similar to general functional identities in that they involve an unknown function. It is important for you to get used to this "higher type variable," an unknown function. This chapter can form a bridge between the specific identities of high school and the unknown function variables from rules of calculus and differential equations.

In high school you learned that trig functions satisfy certain identities or that logarithms have certain "properties." All the the identities you need to recall from high school are

but you must be able to use these identities. Some practice exercises using these familiar identities are given in the text CD Chapter 28 on high school review. The Math Background book has more information.

A general functional identity is an equation that is satisfied by an unknown function (or a number of functions) over its domain. For example, the function

satisfies f[x+y]=2^(x+y)=2^x2^y=f[x]f[y], so eliminating the two middle terms, we see that the function f[x]=2^x satisfies the functional identity

It is important to pay attention to the variable or variables in a functional identity. In order for an equation involving a function to be a functional identity, the equation must be valid for all values of the variables in question. Equation (ExpSum) is satisfied by the function f[x]=2^x for all x and y. For the function f[x]=x, it is true that f[2+2]=f[2]f[2], but

, so f[x]=x does not satisfy functional identity (ExpSum).

Verify that for any positive number, b, the function f[x]=b^x satisfies the functional identity (ExpSum) for all x and y. Is (ExpSum) valid (for all x and y) for the function f[x]=x² or f[x]=x³? Justify your answer.

Define where x is any positive number. Why does this f[x] satisfy the functional identities

and

where x, y, and k are variables. What restrictions should be placed on x and y for the equations to be valid? What is the domain of the logarithm?

Functional identities are a sort of "higher laws of algebra." Observe the notational similarity between the distributive law for multiplication over addition,

and the additive functional identity

Most functions f[x] do not satisfy the additive identity, for example,

The fact that these are not identities means that for some choices of x and y in the domains of the respective functions f[x]=1/x and

, the two sides are not equal. Using the Mathematical Background book, Chapter 2, you can show that the only differentiable functions that do satisfy the additive functional identity are the functions

. In other words, the additive functional identity is nearly equivalent to the distributive law; the only unknown (differentiable) function that satisfies it is multiplication. Other functional identities such as the seven given at the start of this chapter capture the most important features of the functions that satisfy the respective identities. For example, the pair of functions f[x]=1/x and

do not satisfy the addition formula for the sine function, either.

Find values of x and y so that the left and right sides of each of the additive formulas for 1/x and are not equal.
Show that 1/x and also do not satisfy the identity (SinSum), that is,

is false for some choices of x and y in the domains of these functions.

The goal of this chapter is to give you practice at working with familiar functional identities and to extend your thinking to identities in unknown functions.

1) Suppose that f[x] is an unknown function that is known to satisfy (LogProd) (so f[x] behaves "like" , but we don't know if f[x] is ), and suppose that f[0] is a well-defined number (even though we don't specify exactly what f[0] is). Show that this function f[x] must be the zero function, that is, show that f[x]=0 for every x. (Hint: Use the fact that 0*x=0.)
2) Suppose that f[x] is an unknown function that is known to satisfy (LogPower) for all x>0 and all k. Show that f[1] must equal 0, f[1]=0. (Hint: Fix x=1, and try different values of k.)

Let m and b be fixed numbers and define

Verify that if b=0, this function satisfies the functional identity

for all x and that if , f[x] will not satisfy (Mult) for all x (that is, given a nonzero b, there will be at least one x for which (Mult) is not true).
Prove that any function satisfying (Mult) also automatically satisfies the two functional identities

and

for all x and y.
Suppose f[x] is a function that satisfies (Mult) (and for now that is the only thing you know about f[x]). Prove that f[x] must be of the form , for some fixed number m (this is almost obvious).
Prove that a general power function f[x]=mx^k, where k is a positive integer and m is a fixed number, will not satisfy (Mult) for all x if (that is, if , there will be at least one x for which (Mult) is not true).
Prove that does not satisfy the additive identity.
Prove that f[x]=2^x does not satisfy the additive identity.

10.1 Additive Functions

In the early 1800s, Cauchy asked the question: Must a function satisfying

be of the form

? This was not solved until the late 1800s by Hamel. The answer is "No." There are some very strange functions satisfying the additive identity that are not simple linear functions. However, these strange functions are not differentiable.

Chapter 2, Functional Identities from the Mathematical Background book, Foundations of Infinitesimal Calculus, explores this problem in more detail.

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