Questions: For the given functions, find (f ∘ g)(x) and (g ∘ f)(x) and the domain of each. f(x)=6/(1-5x), g(x)=1/x (f ∘ g)(x)= (Simplify your answer. Use integers or fractions for any numbers in the expression.)

Solution Steps

To solve the problem of finding the compositions \((f \circ g)(x)\) and \((g \circ f)(x)\), we need to substitute one function into the other. For \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\). For \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\). After finding these compositions, determine the domain of each by identifying the values of \(x\) that make the expressions undefined.

To find the composition \( (f \circ g)(x) \), we substitute \( g(x) = \frac{1}{x} \) into \( f(x) = \frac{6}{1 - 5x} \):

\[ (f \circ g)(x) = f(g(x)) = f\left(\frac{1}{x}\right) = \frac{6}{1 - 5\left(\frac{1}{x}\right)} = \frac{6}{1 - \frac{5}{x}} = \frac{6x}{x - 5} \]

Next, we find the composition \( (g \circ f)(x) \) by substituting \( f(x) \) into \( g(x) \):

\[ (g \circ f)(x) = g(f(x)) = g\left(\frac{6}{1 - 5x}\right) = \frac{1}{\frac{6}{1 - 5x}} = \frac{1 - 5x}{6} \]

For \( (f \circ g)(x) \), we need to ensure that \( g(x) \) is defined and that \( f(g(x)) \) does not lead to any undefined expressions. The function \( g(x) \) is defined for \( x \neq 0 \), and \( f(g(x)) \) is defined as long as \( 1 - 5\left(\frac{1}{x}\right) \neq 0 \), which simplifies to \( x \neq 5 \). Thus, the domain for \( (f \circ g)(x) \) is:

\[ \text{Domain of } (f \circ g)(x): x \neq 0, x \neq 5 \]

For \( (g \circ f)(x) \), we need \( f(x) \) to be defined and \( g(f(x)) \) to be defined. The function \( f(x) \) is defined for \( x \neq \frac{1}{5} \), and \( g(f(x)) \) is defined as long as \( f(x) \neq 0 \). Since \( f(x) \) is never zero, the domain for \( (g \circ f)(x) \) is:

\[ \text{Domain of } (g \circ f)(x): x \neq \frac{1}{5} \]

Final Answer