Difference between equivalence and equality with functions

[ravern]

I feel aggravatingly close to the answer to this one, but have caved in.

Using ~ for "is equivalent to"

A given question reads:

Given that f(x) ~ 3 - 5x + x^3, show that the equation f(x)=0 has a root x = a, where a lies in the interval 1 < a < 2.

Clearly asking for the answer here would be more than a little insulting. What I want to know is why it is necessary to use equivalence rather than equality at the beginning.

Many thanks.

 

[mathman]

In the example you show, equivalence means is defined as, that is f(x) is defined by the expression. Equality means an actual equation, like find the values of x for which f(x)=0. Here f(x)=0 is an equation, since f(x) is a substitute for the expression for x. f(x) equivalent to 0 would mean f(x) is 0 irrespective of x.

 

[arildno]

You could do it the following way:

The function value for arbitrary x equals the symbol f(x), i.e, f(x)=3-5x+x^3.
You wish to find a particular value (or values) among the possible x's, call that X, so that
f(X)=0, that is solve the following equation for X:
3-5X+X^3=0.

"x" denotes an arbitrary element within the function's domain, "X" denotes those of these such that f(X)=0 is a true statement (for most x's, it is an untrue statement).

 

[HallsofIvy]

I'm not sure I understand arildno's response! The problem I have understanding the whole problem is "equivalent". While there is a single concept of "equal", "equivalent" normally means "the same in some specific way" and you haven't said what that "way" is!

Typically, when you say two things are "equivalent", rather than "equal", you are saying they satisfy some given equivalence relation. Is there mention of an equivalence relation in this?

It's obvious that 3- 5x+ x³ HAS a zero between x= 1 and x= 2 but I don't know what is meant by "equivalent".