Understanding the Domain of a Vector Function
Why the Domain Matters
A vector function gives a position, velocity, or field value through several component functions. Its domain is the set of parameter values that make every component meaningful. This calculator studies that common domain over a chosen interval. It checks the three component expressions first. It then checks optional denominator, radical, and logarithm restrictions.
How This Tool Works
The tool is numerical, so it works best as a careful scanner. It does not replace a formal proof. It helps you find gaps, likely exclusions, and useful interval evidence. You can enter components such as sqrt(t+1), 1/(t-2), and ln(t+3). You should also enter t+1 as a radical condition, t-2 as a denominator condition, and t+3 as a logarithm condition.
Rules Behind the Domain
For a vector function r(t)=<f(t),g(t),h(t)>, the domain is the intersection of the domains of f, g, and h. Denominators cannot equal zero. Even root radicands cannot be negative. Logarithm arguments must be positive. Trigonometric and exponential functions are usually defined for all real input, unless their inner expressions create another restriction.
Reading the Result
The scanned range lets you focus on a practical study window. More samples give finer interval estimates. A smaller tolerance treats values closer to zero as important. Precision controls displayed output only. It does not change the internal scan.
Exports and Study Value
Results appear immediately below the header after submission. The summary shows approximate valid intervals, counts, and sample rows. The table includes component values and invalid reasons. CSV export is useful for spreadsheets. PDF export is useful for reports, assignments, and quick records.
Final Algebra Check
Always read the result as approximate. If a boundary is important, test values very close to that point. Then solve the exact inequalities by hand. For example, sqrt(t+1) requires t >= -1. The expression 1/(t-2) excludes t=2. The expression ln(t+3) requires t > -3. The final domain is the overlap of all such rules. That overlap is the reliable mathematical answer. Use the calculator to organize that reasoning, confirm examples, and prepare clean work.
You can also compare different study windows. This is helpful when a curve has many repeated gaps. Save each run, compare exported rows, and refine the exact domain with algebra before final written submission.