1. **State the problem:** We are given two functions $h(x) = \frac{1}{2x - 8}$ and $g(x) = -2x$. We need to find the domain and range of the composition $(h \circ g)(x) = h(g(x))$. 2. **Write the composition:** $$(h \circ g)(x) = h(g(x)) = h(-2x) = \frac{1}{2(-2x) - 8} = \frac{1}{-4x - 8}$$ 3. **Find the domain:** The domain of $(h \circ g)(x)$ consists of all $x$ values for which the denominator is not zero. Set denominator $\neq 0$: $$-4x - 8 \neq 0$$ $$-4x \neq 8$$ $$x \neq \cancel{-2} \quad \text{(dividing both sides by } -4 \text{ flips inequality but since it's not an inequality, just division)}$$ So the domain is all real numbers except $x = -2$. 4. **Find the range:** The function $(h \circ g)(x) = \frac{1}{-4x - 8}$ is a rational function with a vertical asymptote at $x = -2$ and horizontal asymptote at $y = 0$. Since the numerator is constant 1 and denominator can take any real value except 0, the output can be any real number except 0. Therefore, the range is all real numbers except $0$. **Final answer:** - Domain: $\{x \mid x \neq -2\}$ - Range: $\{y \mid y \neq 0\}$