The functions f(x) and g(x) are defined as:
[tex]\begin{gathered} f(x)\text{ = }\frac{3}{x-7} \\ g(x)\text{ =}\frac{1}{x} \end{gathered}[/tex]
Let us begin by defining a composite function
Function composition is an operation ∘ that takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)).
a) f o g:
[tex]\begin{gathered} (f\text{ o g)(x) = }\frac{3}{\frac{1}{x}\text{ -7}} \\ =\text{ }\frac{3}{\frac{1-7x}{x}} \\ =\text{ }\frac{3x}{1-7x} \end{gathered}[/tex]
The domain is the set of allowable x-values. Since the function is rational, the values of x that would make the function undefined can be obtained by setting the denominator to zero:
[tex]\begin{gathered} 1-\text{ 7x = 0} \\ 7x\text{ = 1} \\ x\text{ = }\frac{1}{7} \end{gathered}[/tex]
Hence, the domain using interval notation is:
[tex](-\infty\text{ , }\frac{1}{7})\text{ U (}\frac{1}{7}\text{ , }\infty)[/tex]
b) g o f:
[tex]\begin{gathered} (g\text{ o f)(x) = }\frac{1}{\frac{3}{x-7}} \\ =\text{ }\frac{x-7}{3} \end{gathered}[/tex]
The domain is:
[tex](-\infty,\text{ }\infty)[/tex]
