1. Let's start by stating the problem: We want to understand the concepts of domain and range in functions. 2. The **domain** of a function is the set of all possible input values (usually $x$) for which the function is defined. 3. The **range** of a function is the set of all possible output values (usually $y$) that the function can produce. 4. Important rules: - The domain depends on restrictions like division by zero or square roots of negative numbers. - The range depends on the behavior of the function and its outputs. 5. For example, consider the function $f(x) = \sqrt{x}$. - The domain is all $x$ such that $x \geq 0$ because square roots of negative numbers are not real. - The range is all $y$ such that $y \geq 0$ because square roots produce non-negative results. 6. Another example: $g(x) = \frac{1}{x}$. - The domain is all real numbers except $x \neq 0$ because division by zero is undefined. - The range is all real numbers except $y \neq 0$ because $\frac{1}{x}$ never equals zero. 7. To find domain and range: - Identify any restrictions on $x$ for the domain. - Analyze the function's output values for the range. This understanding helps in graphing and solving equations involving functions.