Question

Y=1/xCan you please give a graph use color for function, asymptotes, etc.A short table of easy pointsThe domain and rangePlease identify and label the asymptotes (Please work this as if you didn’t have a calculator)

Answer 1

We will have the following:

First, we are given the expression:

`y=(1)/(x)`

Now we will have a table, from x = -10 to x = 10, that is:

`\begin{cases}f(-10)=-(1)/(10)\Rightarrow f(-10)=-0.1 \\ f(-9)=-(1)/(9)\Rightarrow f(-9)=-0.111111111\ldots \\ f(-8)=-(1)/(8)\Rightarrow f(-8)=-0.125 \\ f(-7)=-(1)/(7)\Rightarrow f(-7)=-0.1428571429\ldots \\ f(-6)=-(1)/(6)\Rightarrow f(-6)=-0.166666666\ldots \\ f(-5)=-(1)/(5)\Rightarrow f(-5)=-0.2 \\ f(-4)=-(1)/(4)\Rightarrow f(-4)=-0.25 \\ f(-3)=-(1)/(3)\Rightarrow f(-3)=-0.333333333\ldots \\ f(-2)=-(1)/(2)\Rightarrow f(-2)=-0.5 \\ f(-1)=-(1)/(1)\Rightarrow f(-1)=-1 \\ f(0)=\infty \\ f(1)=(1)/(1)\Rightarrow f(1)=1 \\ f(2)=(1)/(2)\Rightarrow f(2)=0.5 \\ f(3)=(1)/(3)\Rightarrow f(3)=0.333333333\ldots \\ f(4)=(1)/(4)\Rightarrow f(4)=0.25 \\ f(5)=(1)/(5)\Rightarrow f(5)=0.2 \\ f(6)=(1)/(6)\Rightarrow f(6)=0.166666666\ldots \\ f(7)=(1)/(7)\Rightarrow f(7)=1428571429\ldots \\ f(8)=(1)/(8)\Rightarrow f(8)=1.125 \\ f(9)=(1)/(9)\Rightarrow f(9)=0.111111111\ldots \\ f(10)=(1)/(10)\Rightarrow f(10)=0.1\end{cases}`

From this we can see that the function comes from the negative infinite near y = 0 but never 0, at x = 0 the expression grows in the positive axis and negative axis [From the left and right respectively] and goes to infinity, so at x = 0 the expression is undefined; then to the right it comes from an incredibly great number and approaches y = 0 but never touches it, the graph using the points calculated is:

So, there is a vertical asymptote at x = 0, since it is the point in which the expression is undefined:

Now, from this information we can see that the domain and range are:

`\text{Dom}_f=(-\infty,0)\cup(0,\infty)`
`\text{Ran}_f=(-\infty,0)\cup(0,\infty)`