RTCNTC said:
Find the range of y = 1/x algebraically.
To be a bit more precise (perhaps with an eye towards calculus and analysis), let's talk about the function $f : x \mapsto y = \frac{1}{x}$ with domain $\mathbb{R} \setminus \{0\}$ and co-domain $\mathbb{R}$.
This means that $f$ assigns to every non-zero real number $x$ the real number $y = \frac{1}{x}$.
RTCNTC said:
Steps
1. Find the inverse y = 1/x.
Yes, here that works, because $f$ is indeed invertible. In general, you need to determine those real numbers $y$ for which the equation $f(x) = y$ has
at least one nonzero solution $x$, i.e. those $y \in \mathbb{R}$ for which $\frac{1}{x} = y$ has
at least one solution $x \in \mathbb{R} \setminus \{0\}$.
RTCNTC said:
2. Find domain of inverse of y = 1/x.
Yes, for this particular $f$ this is equivalent to what I wrote above.
RTCNTC said:
3. The domain of the inverse is the range of the original function given.
Correct?
Yes, with the remarks above.
For example, can you do the same question for $g : x \mapsto y = \frac{1}{x^2}$, again with domain $\mathbb{R} \setminus \{0\}$ and co-domain $\mathbb{R}$?