The statement "The domain of (fg)(x) consists of the numbers x that are in the domains of both f and g" is FALSE.
Domain is the values of x in the function represented by y=f(x), for which y exists.
THe given statement is "The domain of (fg)(x) consists of the numbers x that are in the domains of both f and g".
Now we assume the [tex]g(x)=x+2[/tex] and [tex]f(x)=\frac{1}{x-6}[/tex]
So here since g(x) is a polynomial function so it exists for all real x.
[tex]f(x)=\frac{1}{x-6}[/tex] does not exists when [tex]x=6[/tex], so the domain of f(x) is given by all real x except 6.
Now,
[tex](fg)(x)=f(g(x))=f(x+2)=\frac{1}{(x+2)-6}=\frac{1}{x-4}[/tex]
So now (fg)(x) does not exists when x=4, the domain of (fg)(x) consists of all real value of x except 4.
But domain of both f(x) and g(x) consists of the value x=4.
Hence the statement is not TRUE universarily.
Thus the given statement about the composition of function is FALSE.
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