Domain & Range

General Principle: Unless specified otherwise, the domain of most polynomial functions is all real numbers, as they do not have inherent mathematical restrictions. However, certain operations impose limitations on the inputs.

Division by Zero: A critical restriction is that division by zero is undefined. Therefore, any input value that would make the denominator of a rational function equal to zero must be excluded from the domain.

Square Roots of Negative Numbers: In the context of real numbers, the square root (or any even root) of a negative number is undefined. Thus, any input value that would result in a negative number under an even root must be excluded from the domain, meaning the expression under the root must be greater than or equal to zero.

Other Restrictions: Logarithmic functions require their arguments to be strictly positive. Trigonometric functions like tangent and secant have domains restricted by values that lead to division by zero (e.g., $\tan(x) = \frac{\sin(x)}{\cos(x)}$ is undefined when $\cos(x) = 0$).

Explicitly Stated Domains: Sometimes, a function's domain is explicitly given as part of its definition, even if the mathematical expression itself would allow for a larger set of inputs. In such cases, the stated domain overrides any implicit restrictions.

Dependence on Domain: The range of a function is directly influenced by its domain. The outputs generated are only those resulting from the allowed inputs.

Analyzing Function Behavior: To find the range, one must consider how the function transforms the values within its domain. This often involves understanding the shape of the function's graph.

Quadratic Functions: For a quadratic function $f(x) = ax^2 + bx + c$, if $a > 0$, the parabola opens upwards, and the minimum value (vertex y-coordinate) determines the lower bound of the range. If $a < 0$, it opens downwards, and the maximum value determines the upper bound.

Rational Functions: The range of rational functions can be complex, often involving horizontal asymptotes or values that the function never reaches. Analyzing limits and graphical behavior is crucial.

Square Root Functions: For $f(x) = \sqrt{g(x)}$, since $\sqrt{\text{non-negative number}}$ always yields a non-negative result, the range will typically be $f(x) \ge 0$ or a subset thereof, depending on the minimum value of $g(x)$ and any vertical shifts.

Using Graphs: Sketching the graph of a function is an effective strategy for visualizing and determining its range. The range corresponds to all the y-values covered by the graph over its defined domain.

Inequality Notation: Domains and ranges are frequently expressed using inequalities. For example, $x \ge 0$ for a domain or $f(x) > 2$ for a range.

Set-Builder Notation: This notation describes the properties of the elements in the set. For example, $\{x \in \mathbb{R} \mid x \ne 0\}$ means 'the set of all real numbers x such that x is not equal to 0'.

Interval Notation: This uses parentheses and brackets to denote intervals. For example, $[0, \infty)$ means $x \ge 0$, and $(-\infty, 0) \cup (0, \infty)$ means $x \ne 0$.

Variable Usage: It is crucial to use 'x' when describing the domain and 'f(x)' or 'y' when describing the range. Confusing these variables is a common error.

Domain vs. Range: The domain refers to the set of all valid inputs (x-values), while the range refers to the set of all possible outputs (f(x) or y-values). They are distinct sets of numbers representing different aspects of a function's behavior.

Determining Domain vs. Range: Determining the domain primarily involves identifying values that cause mathematical impossibilities (like division by zero or even roots of negatives). Determining the range often requires analyzing the function's behavior over its domain, sometimes by sketching its graph or finding extreme values.

Impact of Restrictions: A restricted domain directly limits the possible outputs, thereby affecting the range. For instance, if a function $f(x) = x^2$ has a domain of $x \ge 0$, its range would be $f(x) \ge 0$. However, if its domain were $x \le 0$, its range would still be $f(x) \ge 0$, demonstrating how different domains can lead to the same range.

Graphical Interpretation: On a graph, the domain corresponds to the projection of the function onto the x-axis, indicating all x-coordinates covered by the graph. The range corresponds to the projection of the function onto the y-axis, indicating all y-coordinates covered by the graph.

Identify Restriction Types: When finding the domain, first check for denominators (set to zero and exclude) and even roots (set expression under root $\ge 0$). These are the most common sources of domain restrictions.

Sketch the Graph: For determining the range, especially for non-linear functions, a quick sketch of the graph can be invaluable. It helps visualize the minimum and maximum output values and any asymptotes.

Test Boundary Values: If the domain is an interval, evaluate the function at the endpoints of the interval to understand the behavior of the outputs. Be mindful of whether the endpoints are included or excluded.

Consider Asymptotes: For rational functions, identify vertical asymptotes (from domain restrictions) and horizontal asymptotes, as these often define boundaries for the range.

Check for Symmetry and Extrema: For functions like quadratics, finding the vertex (an extremum) is key to determining the range. Understanding symmetry can also help in visualizing the graph's extent.