Find domain of commposite function f(g(x))

Question

f(x)=1/x+3 and g(x)=- 2/x

Answer 1

Should the original problem contain a parenthesis for grouping so that it

is actually:

 

f(x) = 1/(x+3)

g(x) = -2/x

 

If so we have original stipulations that  x≠-3 from f(x) and x ≠ 0 from g(x)

since we cannot divide by zero.

 

f(g(x)) = 1/((-2/x)+3)

           = 1/[(3x-2)/x]

           = x/(3x-2)

 

The stipulation for f(g(x)) is that x ≠ 2/3 since this would case 3x-2 = 0

 

So our domain, being all possible input values for x, is all real numbers

except -3, 0, 2/3

 

domain = (-∞,-3) U (-3,0) U (0, 2/3) U (2/3, ∞)

Answer 2

For x to be in the domain of f(g(x)), x has to be in the domain of g(x), so that g(x) is defined, and g(x) has to be in the domain of f(x), so that f(g(x) makes sense.  

 

Domain f(g(x)) = {x l x ∈ domg and g(x) ∈ domf}

 

                     = {x l x ≠ 0 and -2/x ≠ -3}

 

                     = {x l x ≠ 0 and x ≠ 2/3}

 

                     = (-∞, 0) ∪ (0, 2/3) ∪ (2/3, ∞)