How do you find the domain and range and intercepTs for #R(x) = (x^2 + x - 12)/(x^2 - 4)#?

1 Answer
Apr 7, 2017

Answer:

see explanation.

Explanation:

To find any #color(blue)"excluded values"# on the domain.

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

#"solve "x^2-4=0rArrx^2=4rArrx=+-2#

#rArr"domain is " x inRR,x!=+-2#

To find #color(blue)"excluded values"# on the range.

divide all terms on numerator/denominator by the highest power of x, that is #x^2#

#R(x)=(x^2/x^2+x/x^2-(12)/x^2)/(x^2/x^2-4/x^2)=(1+1/x-(12)/x^2)/(1-4/x^2)#

as #xto+-oo,1/x,(12)/x^2,4/x^2to0#

#lim_(xto+-oo),R(x)to(1+0-0)/(1-0)#

#=1/1=1larrcolor(red)" excluded value"#

#"range is " y inRR,y!=1#

#color(blue)"intercepts"#

#• " let x = 0, in equation, for y-intercept"#

#• " let y = 0, in equation, for x-intercepts"#

#x=0toy=(-12)/(-4)=3larrcolor(red)" y-intercept"#

The numerator is the only part of R ( x ) that can equal zero when y = 0

#rArry=0tox^2+x-12=0#

#rArr(x+4)(x-3)=0#

#rArrx=-4,x=3larrcolor(red)" x-intercepts"#
graph{(x^2+x-12)/(x^2-4) [-10, 10, -5, 5]}