Question

How do you find the domain of `f(x) = 8 /( x + 4)`?

Answer 1

Domain: `(-oo,-4)uu(-4,oo)`

Explanation:

The domain of a rational function is all the values except for those which that make it undefined. In other words, the denominator of a rational function cannot be `0`.

To find which values exactly make the rational function undefined, we set the denominator not equal to `0` and solve for `x`

`x+4!=0`

`x+cancel(4-4)!=0-4`

`x!=-4`

What we have solved for tells us the values (`x-`values to be precise) that cannot be in the domain.

Therefore our domain is all real numbers except `-4`. In interval notation, this is:

`(-oo,-4)uu(-4,oo)`

What this tells us about the graph as well is that there is a vertical asymptote defined by the line `x=-4` (See graph)

graph{8/(x+4) [-23.37, 16.63, -9.44, 10.56]}

Answer 2

Domain of `f(x) = (-oo, -4)uu (-4, +oo)`

Explanation:

`f(x) = 8/(x+4)`

`f(x)` is defined for all x except `x=-4` where the function is undefined.

`:.` the domain of `f(x) = forall x in RR: x!= -4`

In interval notation: Domain of `f(x) = (-oo, -4)uu (-4, +oo)`

The domain can be seen on the graph of `f(x)` below.

graph{8/(x+4) [-32.47, 32.48, -16.24, 16.23]}