MHB Find the Range of a Rational Function.

  • Thread starter Thread starter mathdad
  • Start date Start date
mathdad
Messages
1,280
Reaction score
0
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?
 
Mathematics news on Phys.org
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 $\mathbb{R}$?
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top