Question

If f(x) is a function of inverse variation and a>b, then is `f(a)>f(b)`?

Answer 1

No; in fact, `f(a) < f(b)`.

Explanation:

When a function is an inverse variation, it means that the function value is inversely proportional to the input. That is, when the input doubles, the output gets cut in half.

Say, for example, that you go for a walk around the block. On your first lap, it only takes you 5 minutes. On your second lap, though, it takes you 10 minutes. What happened to your speed on the second lap? It must have been halved, since the amount of time was twice that of the first lap. This is because velocity and time are inversely proportional, as shown in the equation `v=d/t`. That is...

If `v_1=d/t_1` and `t_2=2t_1`,
then `v_2=d/t_2=d/(2t_1)=1/2(d/t_1)" "=1/2 v_1`

For a fixed distance `d`, when your time `t` doubles, your velocity `v` must get halved.

So!

We are told that `f(x)` is a function of inverse variation, and we have two inputs to give it, `a` and `b`. Since we also know `a>b`, we must then know that `f(a) < f(b)`. If we decrease the input to `f` (by going from `a` to a smaller input `b`), then `f` itself must increase.

Just like if we decrease the amount of time it takes to go around the block, then our velocity (as a function of time) must increase.