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

##### 1 Answer

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

Method 2: Using knowledge of translations

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.

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** .

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.