SOLUTION: Please help. find the domain of the composite of FoG and GoF. a)f(x)=square root of x+1 and g(x)=2x+4/3x-4 b)f(x)=1/x squared and g(x)=|x-1|

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Please help. find the domain of the composite of FoG and GoF. a)f(x)=square root of x+1 and g(x)=2x+4/3x-4 b)f(x)=1/x squared and g(x)=|x-1|       Log On


   

Question 981000: Please help.
find the domain of the composite of FoG and GoF.
a)f(x)=square root of x+1 and g(x)=2x+4/3x-4
b)f(x)=1/x squared and g(x)=|x-1|

Answer by josgarithmetic(39613) About Me  (Show Source):
You can put this solution on YOUR website!
F%3C%3Ef; G%3C%3Eg.

f(x)=sqrt(x+1), g(x)=(2x+4)/(3x-4);
f%28g%28x%29%29=sqrt%28%28%282x%2B4%29%2F%283x-4%29%29%2B1%29

The restrictions for x are that 3x-4%3C%3E0, meaning x%3C%3E4%2F3, and %282x%2B4%29%2F%283x-4%29%2B1%3E=0.

Maybe easier to look at critical values for x. They are -2 and 4%2F3. These cut the x-axis into three intervals:

(-infinity,-2], [-2, 4/3), and (4/3, infinity).

Test ANY value within each interval and find if the interval is part of the domain. Remember, division by 0 is impossible, and square root function's input must be non-negative.

Interval (-infinity,-2]
Test -3.
sqrt%28%28%282%28-3%29%2B4%29%2F%283%28-3%29-4%29%29%2B1%29
sqrt%28%28-2%29%2F%28-13%29%2B1%29
sqrt%282%2F13%2B1%29
acceptable

Interval [-2, 4/3)
Test 1
sqrt%28%28%282%2A1%2B4%29%2F%283%2A1-4%29%29%2B1%29
sqrt%286%2F%28-1%29%2B1%29=sqrt%28-6%2B1%29
NOT acceptable

Interval (4/3, infinity)
Test 2
sqrt%28%28%282%2A2%2B4%29%2F%283%2A2-4%29%29%2B1%29
sqrt%288%2F2%2B1%29
this will work,
Acceptable.

Domain:
The set of real numbers, (-infinity,-2] and (4/3, infinity).