Question

How do you find the domain and asymptotes for `1 / (x+6)`?

Answer 1

Domain x: {x∈R, x≠-6} i.e. x can be all real numbers except x= -6 because there is a vertical asymptote
Horizontal asymptote y=0
Vertical asymptote x= -6

Explanation:

Method 1: Graphing
graph{1/(x+6) [-10, 10, -5, 5]}

It is easy to derive asymptotes from a graph.
As you can see, the graph never touches the x-axis y=0 but only approaches it. This makes y=0 the horizontal asymptote.

Also the graph never touches x= -6, but only approaches it, making x-=-6 the vertical asymptote.

As x = -6 is the only position where x is not possible in the function `y=1/(x+6)`, it must be included in the domain as an exception from all real numbers. Hence the domain is indicated as x: {x∈R, x≠-6}

Method 2: Using knowledge of translations
`y=1/(x+6)`

Finding the horizontal asymptote:
To find this, you need to take note of any vertical shifts in the graph by looking at the equation.
Vertical shifts are added at the back of the function.
`y=1/(x+6)+0`
Since there is no vertical shift in this case, the horizontal asymptote remains at y=0

Finding the vertical asymptote:
To find this, you need to take note of any horizontal shifts in the graph by looking at the equation.
Horizontal shifts are indicated in the brackets, next to x .
`y=1/(x+6)`
In this case, the horizontal shift is +6, which means 6 units to the left (If you're not clear of this, research linear transformations), making the vertical asymptote x=-6

Since the vertical asymptote indicates an impossible x value, it has to be included in the domain as shown above.