Do the functions $f(x)=x^3$ and $g(x)=0$ intersect?

Question

In my math class I was told that a tangent of two functions has two properties:

1) $f(x) = g(x)$ they have a common point.

2) $f'(x) = g'(x)$ at that point the derivatives are equal.

An intersection of two functions also have a common point but their derivatives should not be the same, as $f(x)=x^2$ and $g(x)=0$ then would have an intersection.. which is not true.

So my question is, do $f(x)=x^3$ and $g(x)=0$ intersect? It definitely does cross the graph to the other side but following the rules I stated above it would theoretically not be called an intersection.

Thanks for your thoughts,

Merijn

Answer 1

It really just depends on what you call an "intersection". Do you think that $g(x) = 0$ and $f(x) = x^3$ intersect? Then the definition you gave before is insufficient. In that case, perhaps the following definition is better: $f$ and $g$ intersect at the point $a$ if there are $a_0 < a < a_1$ such that $f(a) = g(a)$, and either $$ f(x) > g(x)\text{ for $x \in (a_0,a)$ and }f(x) < g(x)\text{ for $x \in (a,a_1)$,} $$ or $$ f(x) < g(x)\text{ for $x \in (a_0,a)$ and }f(x) > g(x)\text{ for $x \in (a,a_1)$.} $$ Definitions don't just appear out of nowhere. We use them to quickly describe phenomena that we care about. If we are interested in intersections of graphs, we should make a definition of the word "intersection" that includes all the cases we care about.

Edit: but for the sake of a math class, you should stick to whatever definition they use.

Answer 2

Intersect is understood as "have (at least) a common point", which is true for $x^2$ vs $0$ as well as $x^3$ vs $0$.

Tangency requires a common slope in addition to intersection, and this is also true for $x^2$ vs $0$ and $x^3$ vs $0$ (all slopes being $0$).

Tangency does not exclude intersection, on the opposite it requires it.

Also note that the fact that the curves just touch and stay on the same side instead of cross each other is irrelevant in the definition of tangency.