This shows that the function is undefined at
$y=q$ . Therefore the range of
$f\left(x\right)=\frac{a}{x}+q$ is
$\left\{f\right(x):f(x)\in (-\infty ;q)\cup (q;\infty \left)\right\}$ .
For example, the domain of
$g\left(x\right)=\frac{2}{x}+2$ is
$\{x:x\in \mathbb{R},x\ne 0\}$ because
$g\left(x\right)$ is undefined at
$x=0$ .
We see that
$g\left(x\right)$ is undefined at
$y=2$ . Therefore the range is
$\left\{g\right(x):g(x)\in (-\infty ;2)\cup (2;\infty \left)\right\}$ .
Intercepts
For functions of the form,
$y=\frac{a}{x}+q$ , the intercepts with the
$x$ and
$y$ axis is calculated by setting
$x=0$ for the
$y$ -intercept and by setting
$y=0$ for the
$x$ -intercept.
There are two asymptotes for functions of the form
$y=\frac{a}{x}+q$ . Just a reminder, an asymptote is a straight or curved line, which the graph of a function will approach, but never touch. They are determined by examining the domain and range.
We saw that the function was undefined at
$x=0$ and for
$y=q$ . Therefore the asymptotes are
$x=0$ and
$y=q$ .
For example, the domain of
$g\left(x\right)=\frac{2}{x}+2$ is
$\{x:x\in \mathbb{R},x\ne 0\}$ because
$g\left(x\right)$ is undefined at
$x=0$ . We also see that
$g\left(x\right)$ is undefined at
$y=2$ . Therefore the range is
$\left\{g\right(x):g(x)\in (-\infty ;2)\cup (2;\infty \left)\right\}$ .
From this we deduce that the asymptotes are at
$x=0$ and
$y=2$ .
Sketching graphs of the form
$f\left(x\right)=\frac{a}{x}+q$
In order to sketch graphs of functions of the form,
$f\left(x\right)=\frac{a}{x}+q$ , we need to determine four characteristics:
domain and range
asymptotes
$y$ -intercept
$x$ -intercept
For example, sketch the graph of
$g\left(x\right)=\frac{2}{x}+2$ . Mark the intercepts and asymptotes.
We have determined the domain to be
$\{x:x\in \mathbb{R},x\ne 0\}$ and the range to be
$\left\{g\right(x):g(x)\in (-\infty ;2)\cup (2;\infty \left)\right\}$ . Therefore the asymptotes are at
$x=0$ and
$y=2$ .
There is no
$y$ -intercept and the
$x$ -intercept is
${x}_{int}=-1$ .
Draw the graph of
$y=\frac{-4}{x}+7$ .
The domain is:
$\{x:x\in \mathbb{R},x\ne 0\}$ and the range is:
$\left\{f\right(x):f(x)\in (-\infty ;7)\cup (7;\infty \left)\right\}$ .
We look at the domain and range to determine where the asymptotes lie. From the domain we see that the function is undefined when
$x=0$ , so there is one asymptote at
$x=0$ . The other asymptote is found from the range. The function is undefined at
$y=q$ and so the second asymptote is at
$y=7$
There is no y-intercept for graphs of this form.
The x-intercept occurs when
$y=0$ . Calculating the x-intercept gives:
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?