Questions: Refer to functions (q) and (r). Evaluate ((q circ r)(x)) and write the domain in interval notation. Write your answers as integers or simplified fractions [ q(x)=frac1x-4 quad r(x)=2 x+3 ] [ (q circ r)(x)= ]

Solution Steps

We have the functions defined as follows: \[ q(x) = \frac{1}{x - 4} \] \[ r(x) = |2x + 3| \]

To find \((q \circ r)(x)\), we substitute \(r(x)\) into \(q(x)\): \[ (q \circ r)(x) = q(r(x)) = q(|2x + 3|) = \frac{1}{|2x + 3| - 4} \]

The function \((q \circ r)(x)\) is defined as long as the denominator is not zero: \[ |2x + 3| - 4 \neq 0 \] This leads to: \[ |2x + 3| \neq 4 \]

We solve the equation \( |2x + 3| = 4 \):

1. \(2x + 3 = 4\) gives \(2x = 1\) or \(x = \frac{1}{2}\).
2. \(2x + 3 = -4\) gives \(2x = -7\) or \(x = -\frac{7}{2}\).

Thus, the values \(x = \frac{1}{2}\) and \(x = -\frac{7}{2}\) are not in the domain.

The domain excludes the points found above: \[ \text{Domain} = (-\infty, -\frac{7}{2}) \cup (-\frac{7}{2}, \frac{1}{2}) \cup (\frac{1}{2}, \infty) \]

Final Answer