Welcome to finding domain and range! The domain of a function consists of all possible input values, or x-values, that we can substitute into the function. The range consists of all possible output values, or y-values, that the function can produce. Let's visualize this with a simple quadratic function f of x equals x squared.
When finding the domain, we must identify values that make the function undefined. The most common restrictions are division by zero, negative values under square roots, non-positive arguments in logarithms, and specific ranges for inverse trigonometric functions. For example, the function f of x equals one over x minus two is undefined when x equals two, since this makes the denominator zero. Therefore, the domain excludes x equals two.
Finding the range requires analyzing the function's output behavior. We look for minimum and maximum values, consider the function's end behavior, and identify any horizontal asymptotes. For the quadratic function f of x equals x squared plus one, the vertex occurs at zero comma one, which gives us the minimum output value. Since the parabola opens upward, the range starts at one and extends to positive infinity.
Let's work through a specific example: the square root function f of x equals square root of x minus three. For the domain, we need the expression under the square root to be non-negative, so x minus three must be greater than or equal to zero, which means x is greater than or equal to three. For the range, since square roots always produce non-negative outputs, the range starts at zero and extends to positive infinity.
To summarize, finding domain and range follows systematic steps. For domain, start with all real numbers and identify restrictions like division by zero or negative square roots. For range, analyze the function's behavior, find minimum and maximum values, and consider end behavior. Different function types have characteristic domain and range patterns. Linear functions typically have domain and range of all real numbers, quadratics have restricted ranges, and rational functions often exclude specific values from both domain and range. Practice with various function types to master these concepts.