Questions: Find the domain of the composite functions (f circ g) and (g circ f). (f(x)=sqrt3-x) ; (g(x)=frac1x-6) What is the domain of the composite function (g circ f) ? Domain: (square) (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

Solution Steps

To find the domain of the composite functions \( f \circ g \) and \( g \circ f \), we need to consider the domains of the individual functions and how they interact when composed.

1. Domain of \( f \circ g \):

   • First, determine the domain of \( g(x) = \frac{1}{x-6} \), which is all real numbers except \( x = 6 \).
   • Then, find the domain of \( f(x) = \sqrt{3-x} \), which is \( x \leq 3 \).
   • For \( f \circ g \), \( g(x) \) must be in the domain of \( f \). Therefore, solve \( 3 - \frac{1}{x-6} \geq 0 \).

2. Domain of \( g \circ f \):

   • First, determine the domain of \( f(x) = \sqrt{3-x} \), which is \( x \leq 3 \).
   • Then, find the domain of \( g(x) = \frac{1}{x-6} \), which is all real numbers except \( x = 6 \).
   • For \( g \circ f \), \( f(x) \) must be in the domain of \( g \). Therefore, solve \( \sqrt{3-x} \neq 6 \).

To find the domain of the composite function \( f \circ g \), we start with the function \( g(x) = \frac{1}{x - 6} \). The domain of \( g \) is all real numbers except \( x = 6 \). Next, we need to ensure that \( g(x) \) is within the domain of \( f(x) = \sqrt{3 - x} \), which requires \( 3 - g(x) \geq 0 \).

Solving the inequality: \[ 3 - \frac{1}{x - 6} \geq 0 \] leads to the condition: \[ x \geq \frac{19}{3} \quad \text{or} \quad x < 6 \]

Thus, the domain of \( f \circ g \) is: \[ \left(-\infty, 6\right) \cup \left[\frac{19}{3}, \infty\right) \]

Next, we find the domain of the composite function \( g \circ f \). The function \( f(x) = \sqrt{3 - x} \) has a domain of \( x \leq 3 \). For \( g(f(x)) \) to be defined, we need \( f(x) \neq 6 \).

Setting up the equation: \[ \sqrt{3 - x} \neq 6 \] This leads to: \[ 3 - x \neq 36 \quad \Rightarrow \quad x \neq -33 \]

Since \( x \) must also satisfy \( x \leq 3 \), the domain of \( g \circ f \) is: \[ (-\infty, 3] \]

Final Answer