Test questions on finding domain and range of radical expressions

📚 Quick Study Guide

• 🔍 Radical Expressions: These are expressions containing a radical symbol ($\sqrt{}$). The most common type is the square root.
• 💡 Domain: The domain of a radical expression is the set of all real numbers that, when substituted for the variable, result in a real number. For square roots, this means the expression inside the radical (the radicand) must be greater than or equal to zero.
• 📝 Finding the Domain: To find the domain of $\sqrt{f(x)}$, solve the inequality $f(x) \geq 0$.
• 📈 Range: The range is the set of all possible output values (y-values) of the radical expression. For square roots, the range is typically all non-negative real numbers (i.e., $y \geq 0$), unless there are transformations applied to the radical.
• 🧪 Transformations: Transformations such as vertical shifts (adding or subtracting a constant outside the radical) or vertical reflections (multiplying by -1 outside the radical) can affect the range.
• 🔢 Example: For the expression $\sqrt{x-2}$, the domain is $x \geq 2$ and the range is $y \geq 0$.

Practice Quiz

1. What is the domain of the function $f(x) = \sqrt{x + 5}$?
   1. $x \geq 5$
   2. $x \leq -5$
   3. $x \geq -5$
   4. $x \leq 5$
2. What is the domain of the function $g(x) = \sqrt{2x - 6}$?
   1. $x \geq -3$
   2. $x \geq 3$
   3. $x \leq 3$
   4. $x \leq -3$
3. What is the range of the function $h(x) = \sqrt{x} + 3$?
   1. $y \geq 0$
   2. $y \geq 3$
   3. $y \leq 3$
   4. All real numbers
4. What is the domain of the function $k(x) = \sqrt{7 - x}$?
   1. $x \geq 7$
   2. $x \leq 7$
   3. $x \geq -7$
   4. $x \leq -7$
5. What is the range of the function $m(x) = -\sqrt{x}$?
   1. $y \geq 0$
   2. $y \leq 0$
   3. All real numbers
   4. $y \geq -1$
6. What is the domain of the function $p(x) = \sqrt{3x + 9}$?
   1. $x \geq 3$
   2. $x \geq -3$
   3. $x \leq 3$
   4. $x \leq -3$
7. What is the range of the function $q(x) = \sqrt{x - 4} + 2$?
   1. $y \geq 0$
   2. $y \geq 2$
   3. $y \geq 4$
   4. All real numbers