Question

How do you find the domain of `f(x) = (x+18 )/( x^2-169)`?

Answer 1

`x inRR,x!=+-13`

Explanation:

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be.

`"solve "x^2-169=0rArr(x-13)(x+13)=0`

`rArrx=+-13larrcolor(red)" excluded values"`

`rArr"domain is "x inRR,x!=+-13`

Answer 2

The domain is `x in (-oo,-13) uu(-13,+13)uu(+13,+oo)`

Explanation:

The denominator must `!=0`

Therefore,

`x^2-169!=0`

`(x+13)(x-13)!=0`

Let `g(x)=(x+13)(x-13)`

Construct a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaa)` `-oo` `color(white)(aaaaa)` `-13` `color(white)(aaaaaaa)` `13` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `x+13` `color(white)(aaaaaaa)` `-` `color(white)(aaa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaaaa)` `+`

`color(white)(aaaa)` `x-13` `color(white)(aaaaaaa)` `-` `color(white)(aaa)` ###color(white)(aaaaa)#`-` `color(white)(aa)` `0` `color(white)(aaa)` `+`

`color(white)(aaaa)` `g(x)` `color(white)(aaaaaaaaa)` `+` `color(white)(aaa)` `0` `color(white)(aaaa)` `-` `color(white)(aa)` `0` `color(white)(aaa)` `+`

Therefore,

`g(x)!=0` when `x in (-oo,-13) uu(-13,+13)uu(+13,+oo)`

This is the domain of `f(x)`

graph{(x+18)/(x^2-169) [-20.27, 20.28, -10.14, 10.14]}