How do you find the domain of [fog](x) given f(x)=sqrt(x-2) and g(x)=1/(4x)?
1 Answer
The domain of
Explanation:
First, find
So:
f(color(blue)x)=sqrt(color(blue)x-2)
f(color(blue)(g(x)))=sqrt(color(blue)(g(x))-2)=sqrt(1/(4x)-2)
Simplifying the fraction in the square root:
f(g(x))=sqrt((1-8x)/(4x))
We have a square root function. Note that when there is a square root, the contents of the square root have to be positive, or greater than
(1-8x)/(4x)>0
There are multiple places to analyze here. The numerator is
The denominator is
We need three intervals--one when
Test the sign of
(1-8(-1))/(4(-1))=(1+9)/(-4)=-5/2
Since this is
Testing the sign at
(1-8(1/16))/(4(1/16))=(1-1/2)/(1/4)=2
This is positive. Thus
Testing
(1-8(1))/(4(1))=-7/4
So
The domain of