Algebra question: Will the constants be equal?

Question

If:

$t^3+e^t+t^{-1/2}+C_1=f(t)+C_2$

where $C_1$ and $C_2$ are constants

$f(t)$ is a function of $t$ without any constant term

then will $C_1=C_2?$

Answer 1

There's no such thing as "a function without any constant term"; whether $f$ "has" a constant term depends on how $f$ is written, not on $f$ itself. You can talk about, for example, polynomials with no constant term; that means there is no constant term in the standard expression for the polynomial.

Honest. For example, say $f(t)=\sin^2(t)$. Does $f$ have a constant term? I imagine you'd say no. Now let $g(t)=1-\cos^2(t)$; I imagine you'd say yes, $g$ does have a constant term.

But $f$ and $g$ are exactly the same function! Saying "$g$ has a constant term but $f$ does not" is the same as saying "$f$ has a constant term but $f$ does not have a constant term".