1. **Problem:** Find the domain and range of the function $y = x + 4$. 2. **Formula and rules:** For linear functions $y = mx + b$, the domain is all real numbers because $x$ can be any value. 3. **Domain:** $\text{Domain} = (-\infty, \infty)$. 4. **Range:** Since $y$ is linear and unbounded, $\text{Range} = (-\infty, \infty)$. --- 1. **Problem:** Find the domain and range of the function $y = |x|$. 2. **Formula and rules:** The absolute value function outputs non-negative values. 3. **Domain:** $\text{Domain} = (-\infty, \infty)$ because $x$ can be any real number. 4. **Range:** Since $y = |x| \geq 0$, $\text{Range} = [0, \infty)$. --- 1. **Problem:** Find the domain and range of the function $x = |y|^2$. 2. **Rewrite:** $x = (|y|)^2 = y^2$ since squaring removes the absolute value. 3. **Domain:** $y$ can be any real number, so $\text{Domain} = (-\infty, \infty)$. 4. **Range:** Since $x = y^2 \geq 0$, $\text{Range} = [0, \infty)$. --- 1. **Problem:** Find the domain and range of the function $y = -2(x+4)^2 - 1$. 2. **Formula and rules:** This is a quadratic function with vertex at $x = -4$. 3. **Domain:** Quadratic functions have domain $(-\infty, \infty)$. 4. **Range:** Since the coefficient of $(x+4)^2$ is negative, the parabola opens downward. 5. **Vertex value:** $y_{max} = -2(0)^2 - 1 = -1$. 6. **Range:** $\text{Range} = (-\infty, -1]$.