Question f6298

Feb 16, 2018

Range = all real s except 0

Explanation:

Range is, in common tongue, defined as all possible outputs your function can have, given a set of inputs (your domain).

When asked what a function's range is, however, this question is not particularly useful: it's not efficient (or possible) to check every single input and see if it gives you a valid output. Therefore, a more useful question is what outputs could you NOT have for your given domain?

Given the particular function $y = \frac{2}{x}$, The only thing I'm worried about $f \left(x\right)$ being zero. After all, it seems to me that there's no easy way to get $f \left(x\right) = 0$ without plugging in $0$ itself, which is not allowed since we can't divide by 0.

We can test this by setting up an equation:

$\frac{2}{x} = 0$

Multiply both sides by $x$:

$2 = 0$

This is clearly wrong. Hence, it is clear that there's no value of $x$ that could possibly make $f \left(x\right) = 0$. Hence, we can say that the range of $f \left(x\right)$ is all real numbers except 0.

This is also particularly evident given the graph of $f \left(x\right)$:

graph{2/x [-10, 10, -5, 5]}

Notice how $f \left(x\right)$ goes up and down to infinity (meaning that it can be any positive or negative number), but has an asymptote at $y = 0$, meaning that it can get awfully close, but will never equal $y = 0$.

Hope that helped :)