Question

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

Answer 1

Answer:

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

Step-by-step explanation:

we have

$f(x)=\frac{1}{x+3}$

$g(x)=\frac{6}{x+4}$

we know that

$f(g(x))=\frac{1}{(\frac{6}{x+4})+3}$

$f(g(x))=\frac{1}{\frac{6+3(x+4)}{x+4}}$

$f(g(x))=\frac{x+4}{6+3x+12}$

$f(g(x))=\frac{x+4}{3x+18}$

$f(g(x))=\frac{x+4}{3(x+6)}$

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,∞)