MHB Find the Range of a Rational Function.

[mathdad]

Find the range of y = 1/x algebraically.

Steps

1. Find the inverse y = 1/x.

2. Find domain of inverse of y = 1/x.

3. The domain of the inverse is the range of the original function given.

Correct?

 

[S.G. Janssens]

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 $y = \frac{1}{x}$.

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 $x \in \mathbb{R} \setminus \{0\}$.

2. Find domain of inverse of y = 1/x.

Yes, for this particular $f$ this is equivalent to what I wrote above.

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 $\mathbb{R}$?