Determine the domain of the function left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis where lower f left-parenthesis x right-parenthesis equals Start Fraction 3 x minus 1 over x minus 4 End Fraction and lower g left-parenthesis x right-parenthesis equals Start Fraction x plus 1 over x End Fraction.
To determine the domain of the function, we need to find the values of x that are valid inputs for both functions f(x) and g(x).
For f(x), the only value of x that is not allowed is x = 4, since it would make the denominator of the fraction zero.
For g(x), the only value of x that is not allowed is x = 0, since it would make the denominator of the fraction zero.
Therefore, the domain of the function f(g(x)) is all real numbers except x = 0 and x = 4.
Domain: x ≠ 0, x ≠ 4
To determine the domain of the function f(g(x)), we need to consider the domains of both f and g separately and find the intersection of their domains.
For f(x) = (3x - 1)/(x - 4), the only restriction is that the denominator (x - 4) cannot be equal to zero, as division by zero is undefined. So, x ≠ 4.
For g(x) = (x + 1)/x, the denominator (x) cannot be equal to zero, as division by zero is undefined. So, x ≠ 0.
To find the domain of f(g(x)), we need to find the intersection of the domains of f and g. Since both f and g have the restriction x ≠ 0, and f(x) has a restriction x ≠ 4, the intersection is x ≠ 0 and x ≠ 4.
Therefore, the domain of f(g(x)) is all real numbers except x = 0 and x = 4.