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

Question

Julia Adams · Accepted Answer

#x inRR,x!=+-13#

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#

Avery Alexander · Answer

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

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]}

Savannah Anderson · Answer

undefined Domain: $x \in \mathbb{R}$, $x \neq -13, 13$