SOLUTION: 2. Find the domain, range, and asymptotes (if any) of the following functions: a) 𝑓(𝑥) = 4𝑥+3/𝑥+2 b) 𝑓(𝑥) =𝑥+2/𝑥^2−1 c)𝑓(𝑥)=𝑥2−4𝑥+5/𝑥

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: 2. Find the domain, range, and asymptotes (if any) of the following functions: a) 𝑓(𝑥) = 4𝑥+3/𝑥+2 b) 𝑓(𝑥) =𝑥+2/𝑥^2−1 c)𝑓(𝑥)=𝑥2−4𝑥+5/𝑥      Log On


   

Question 1194412: 2. Find the domain, range, and asymptotes (if any) of the following functions:
a) 𝑓(𝑥) = 4𝑥+3/𝑥+2
b) 𝑓(𝑥) =𝑥+2/𝑥^2−1
c)𝑓(𝑥)=𝑥2−4𝑥+5/𝑥−3
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

assuming you have:
a)
f%28x%29+=+%284x%2B3%29%2F%28x%2B2%29
domain:
x%3C-2+or x%3E-2
Interval Notation:
(-infinity,-2) U (-2,infinity)
range:
range of given function is equat to domain of inverse
so, find inverse of %284x%2B3%29%2F%28x%2B2%29 which is %283-2x%29%2F%28x-4%29+ =>domain is x%3C4+or x%3E4
then we say the range of (4x+3)/(x+2) is:
f%28x%29%3C4 or+f+%28x+%29%3E4
asympthotes:
Vertical: x=-2
Horizontal: y=4

b)
f%28x%29+=%28x%2B2%29%2F%28x%5E2-1%29
domain:
x%3C-1+or -1%3Cx%3C1 or x%3E1
Interval Notation:
(-infinity,-1) U (-1,1) U (1,infinity)
range:
f%28x%29+=y
y=%28x%2B2%29%2F%28x%5E2-1%29
y%28x%5E2-1%29=x%2B2
yx%5E2-y-x-2=0
yx%5E2-x-%28y%2B2%29=0
range is where discriminant is >=0
discriminant is b%5E2-4ac
in your case a=y, b=-1,+c=-%28y%2B2%29
%28-1%29%5E2-4y%28-%28y%2B2%29%29%3E=0
1%2B4y%5E2%2B8y%3E=0
4y%5E2%2B8y%2B1%3E=0
y%3E=1%2F2+%28sqrt%283%29+-+2%29=> y%3E=sqrt%283%29+-+1
y%3C=1%2F2+%28-2+-+sqrt%283%29%29+=> y%3C=-1+-+sqrt%283%29%2F2
f%28x%29%3C=+-sqrt%283%29%2F2-1 or f%28x%29%3E=sqrt%283%29%2F2-1
asympthotes:
Vertical: x=-1,x=1
Horizontal: y=0
c)
f%28x%29=%28x%5E2-4x%2B5%29%2F%28x-3%29
domain:
x%3C3+or x%3E3
range:
inverse is %28x+%2B+4%29%2F2+%C2%B1+1%2F2+sqrt%28x%5E2+-+4+x+-+4%29
=>discriminant x%5E2+-+4+x+-+4%3E=0
x%3E=2+%2B+2+sqrt%282%29
x%3C=2+-+2+sqrt%282%29

since domain of inverse is the range of original function,
range is: f%28x%29+%3C=2-2+sqrt%282%29+ or y%3E=2+%2B+2sqrt%282%29

asymptotes:
Vertical: x=3
Horizontal: y=x-1 (slant)