SOLUTION: Find the domain and range of the function f(x) = (x^2 + 1)/( 4x^2 - 8x + 3).

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Question 1183953: Find the domain and range of the function
f(x) = (x^2 + 1)/( 4x^2 - 8x + 3).
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29+=+%28x%5E2+%2B+1%29%2F%28+4x%5E2+-+8x+%2B+3%29
domain:
exclude x that makes denominator equal to zero
4x%5E2+-+8x+%2B+3=0+.......use quadratic formula
x=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a
x=%28-%28-8%29%2B-sqrt%28%28-8%29%5E2-4%2A4%2A3%29%29%2F%282%2A4%29
x=%288%2B-sqrt%2864-48%29%29%2F8
x=%288%2B-sqrt%2816%29%29%2F8
x=%288%2B-4%29%2F8
x=%282%2B-1%29%2F2

solutions
x=%282-1%29%2F2=>x=+1%2F2
x=%282%2B1%29%2F2=>x=+3%2F2
so, domain is:
{ x element R :+x%3C%3E1%2F2 ,x%3C%3E3%2F2+, x%3E3%2F2 }

Interval notation:
(-infinity , 1%2F2 ) U ( 1%2F2, 3%2F2 ) U ( 3%2F2, infinity+)

range:
the range is the set of y for which the discriminant is greater or equal to zero
y=+%28x%5E2+%2B+1%29%2F%28+4x%5E2+-+8x+%2B+3%29
y%28+4x%5E2-8x+%2B+3%29=+x%5E2+%2B+1
4yx%5E2-x%5E2-+8yx+%2B+3y-1=0
%284y-1%29x%5E2-+8yx+%2B+%283y-1%29=0
->discriminant >=0
b%5E2-4ac=%28-8y%29%5E2-4%284y-1%29%283y-1%29
b%5E2-4ac=64y%5E2-48+y%5E2+%2B+28+y+-+4
b%5E2-4ac=16y%5E2+%2B+28+y+-+4
16y%5E2+%2B+28+y+-+4+%3E=0 .......divide by 4
4y%5E2+%2B+7y+-+1+%3E=0
use quadratic formula
y=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29
y=%28-7%2B-sqrt%287%5E2-4%2A4%28-1%29%29%29%2F%282%2A4%29
y=%28-7%2B-sqrt%2849%2B16%29%29%2F8
y=%28-7%2B-sqrt%2865%29%29%2F8

solutions
%28sqrt%2865%29-7%29%2F8%3E=0+
%28-7-sqrt%2865%29%29%2F8%3E=0+

range is:
{ f+%28x+%29 element R : f+%28x+%29+%3C=%28-7-sqrt%2865%29%29%2F8 or +f+%28x+%29+%3E=%28sqrt%2865%29-7%29%2F8 }