Graph $\ell(x) = 3x +2$. What would $\ell(x+1) - \ell(x)$ be for all $x$?

Question

What does it mean to give an answer "for all $x$"? Is it asking for the domain?

Answer 1

The question is asking you to find a result true over all the domain. $\ell(x+1) - \ell(x) = 3(x+1) + 2 - 3x - 2 = 3x + 3 + 2 - 3x - 2 = 3$

This is true over all x because the function $\ell(x)$ is defined for all x, if we are discussing the real numbers.

If you graph both $\ell(x+1)$ and $\ell(x)$ you'll notice that the perpendicular distance between the two lines is always constant.