Questions: For the given functions, find (f ∘ g)(x) and (g ∘ f)(x) and the domain of each. f(x)=(1-x)/(21x), g(x)=1/(1+21x)

Solution Steps

To find 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 the compositions, determine the domain of each by identifying the values of \(x\) that make the denominator zero or cause any other undefined behavior.

To find \((f \circ g)(x)\), substitute \(g(x) = \frac{1}{1 + 21x}\) into \(f(x) = \frac{1-x}{21x}\):

\[ f(g(x)) = f\left(\frac{1}{1 + 21x}\right) = \frac{1 - \frac{1}{1 + 21x}}{21 \cdot \frac{1}{1 + 21x}} \]

Simplifying the expression:

\[ f(g(x)) = \frac{\left(1 - \frac{1}{1 + 21x}\right) \cdot (1 + 21x)}{21} = x \]

To find \((g \circ f)(x)\), substitute \(f(x) = \frac{1-x}{21x}\) into \(g(x) = \frac{1}{1 + 21x}\):

\[ g(f(x)) = g\left(\frac{1-x}{21x}\right) = \frac{1}{1 + 21 \cdot \frac{1-x}{21x}} \]

Simplifying the expression:

\[ g(f(x)) = \frac{1}{1 + \frac{21 - 21x}{21x}} = \frac{1}{\frac{21x + 21 - 21x}{21x}} = x \]

The domain of \((f \circ g)(x)\) is determined by the domain of \(g(x)\), which is all \(x\) except where the denominator is zero:

\[ 1 + 21x \neq 0 \implies x \neq -\frac{1}{21} \]

The domain of \((g \circ f)(x)\) is determined by the domain of \(f(x)\), which is all \(x\) except where the denominator is zero:

\[ 21x \neq 0 \implies x \neq 0 \]

Final Answer