SOLUTION: , find the Domain and the Range of the following funcions. a. f(x) = 2/(5x +6) b. f(x) = 1/√(2x -3) c. f (x) = 1/(x^2-2x -3)

Algebra ->  Sequences-and-series -> SOLUTION: , find the Domain and the Range of the following funcions. a. f(x) = 2/(5x +6) b. f(x) = 1/√(2x -3) c. f (x) = 1/(x^2-2x -3)       Log On


   

Question 1183577: , find the Domain and the Range of the following funcions.
a. f(x) = 2/(5x +6)
b. f(x) = 1/√(2x -3)
c. f (x) = 1/(x^2-2x -3)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
find the Domain and the Range of the following functions.
a. f%28x%29+=+2%2F%285x+%2B6%29
note that %285x+%2B6%29 is denominator and cannot be equal to zero
so exclude vale of x that makes %285x+%2B6%29=0
5x=-6
x=-6%2F5
Domain:
{ x element R : x%3C%3E-6%2F5 }
Range:
{ y element R+: y%3C%3E0 }

+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+2%2F%285x+%2B6%29%29+

b. f%28x%29+=+1%2Fsqrt%282x+-3%29
sqrt%282x+-3%29 must be greater than zero
sqrt%282x+-3%29%3E0 only if 2x+-3%3E0 => 2x+%3E3 => x+%3E3%2F2
Domain:
{ x element R : x%3E3%2F2 }
Range:
{ y element R : y%3E0} (all positive real numbers)

+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C++1%2Fsqrt%282x+-3%29%29+


c. f+%28x%29+=+1%2F%28x%5E2-2x+-3%29....factor denominator
f+%28x%29+=+1%2F%28x%5E2%2Bx-3x+-3%29
f+%28x%29+=+1%2F%28%28x%5E2%2Bx%29-%283x+%2B3%29%29
f+%28x%29+=+1%2F%28x%28x%2B1%29-3%28x+%2B1%29%29
f+%28x%29+=+1%2F%28%28x-3%29%28x+%2B1%29%29
exclude values x=3 and x=-1
Domain:
{ x element R : x%3C%3E-1 and x%3C%3E3 }
Interval notation:
(-infinity ,-1) U (-1,3) U (3,infinity )
Range:
The range is the set of y values of the maximum and minimum
x coordinate of max is
-b%2F2a=-%28-2%29%2F%282%2A1%29=2%2F2=1
x=1
plug it in
f+%281%29+=+1%2F%281%5E2-2%2A1+-3%29
f+%281%29+=+1%2F%281-5%29
f+%281%29+=+-1%2F4
so vertex is at 1, -1%2F4+

minimum at x=3 and x=-1, so y%3E0

range:
{ y element R : y%3C=-1%2F4 or y%3E0 }
+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C++1%2F%28x%5E2-2x+-3%29%29+