How do you identify all asymptotes or holes and intercepts for f(x)=(2x^2+3)/(x^2+6x+8)f(x)=2x2+3x2+6x+8?
1 Answer
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 and if the numerator is non-zero for these values then they are vertical asymptotes.
" solve "x^2+6x+8=0rArr(x+2)(x+4)=0 solve x2+6x+8=0⇒(x+2)(x+4)=0
rArrx=-4" and "x=-2" are the asymptotes"⇒x=−4 and x=−2 are the asymptotes
"horizontal asymptotes occur as"horizontal asymptotes occur as
lim_(xto+-oo),f(x)toc" ( a constant)"
"divide terms on numerator/denominator by the highest"
"power of x that is "x^2
f(x)=((2x^2)/x^2+3/x^2)/(x^2/x^2+(6x)/x^2+8/x^2)=(2+3/x^2)/(1+6/x+8/x^2)
"as "xto+-oo,f(x)to(2+0)/(1+0+0)
rArry=2" is the asymptote"
"holes occur if a common factor is eliminated from"
"the numerator/denominator. This is not the case here"
"hence there are no holes"
color(blue)"Intercepts"
• " let x = 0 for y-intercept"
• " let y = 0 for x-intercepts"
x=0toy=3/8larrcolor(red)"y-intercept"
y=0to2x^2+3=0rArrx^2=-3/2
"this has no real solutions hence no x-intercepts"
graph{(2x^2+3)/(x^2+6x+8) [-10, 10, -5, 5]}
