the domain will be, the values "x" can safely take on.
now, the 0.3x has an "x" with no constraints per se. Let's inspect the parenthesized pair.
let's say x = 32, that makes the fraction 32/32, or 1, that makes that expression 1-1 = 0, so x = 32 is valid in the domain, and when that happens f(x) is 0.
put in other words, when the distance of both ends in the spring is 32, f(x) is 0, namely the spring is not hanging out at all, is just rigid straight.
now, what if the x = 0? the fraction becomes 0/32, or 0, and the expression becomes 0 - 1 = -1, BUT 0.3x becomes 0.3(0) and then f(x) is 0 again.
we can't give "x" values that are negative, since negative values will give us a negative f(x), which for this case makes no sense.
f(x) is the distance the spring is hanging, and a spring can't be hanging at -3 or such.
and we can't give it values higher than 32, say for example we can use x = 64, that gives us f(x) of 19.2, but what if we give it x = 640, that gives us an f(x) of 192, what about x = 640000 and so on, so, I mean, the spring is not infinite, so chances are is 32 inches long, so the constraints for "x" and therefore the domain are between 0 and 32.
therefore, any values that "x" can safely take on, will be from 0( not hanging) to 32(not hanging again), and any values in between, will simply give us the distance it's hanging from the van.
domain is then [0, 32].
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