Question

Given f(x)=1/x+3 and g(x)=6/x+4, find the domain of f(g(x))

Answer 1

The domain is all real numbers except the value of x=-6

Explanation:

we have

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

`g(x)=(6)/(x+4)`

we know that

`f(g(x))=(1)/(((6)/(x+4))+3)`

`f(g(x))=(1)/((6+3(x+4))/(x+4))`

`f(g(x))=(x+4)/(6+3x+12)`

`f(g(x))=(x+4)/(3x+18)`

`f(g(x))=(x+4)/(3(x+6))`

Remember that the denominator cannot be equal to zero

so

The value of x cannot be equal to -6

therefore

The domain is all real numbers except the value of x=-6

In interval notation the domain is

(-∞,-6) ∪ (-6,∞)