Plot functions, test inequalities, and inspect behavior quickly. Compare roots, asymptotes, and solution intervals precisely. Download tables and reports for classroom or homework use.
| Example Item | Value |
|---|---|
| Numerator | x² - 5x + 6 |
| Denominator | x - 4 |
| Inequality | f(x) ≥ 0 |
| Domain | -2 to 8 |
| Sample Step | 0.25 |
| Estimated Satisfying Intervals | [-2, 2] ∪ [3, 3.75] ∪ [4.25, 8] |
| Vertical Restriction | x = 4 |
Polynomial form: P(x) = a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀
Rational form: f(x) = P(x) / Q(x), where Q(x) ≠ 0
Inequality test: Check whether f(x) is greater than, greater than or equal to, less than, or less than or equal to zero.
Graphing method: The calculator samples x-values across the selected domain, evaluates the function, and plots each valid point on the graph.
Root and restriction scan: Numerator roots estimate x-intercepts. Denominator roots estimate vertical restrictions for rational expressions.
Enter the numerator coefficients from x⁴ to the constant term.
Enter denominator coefficients for a rational function. Leave them all as zero to treat the function like a polynomial.
Select the inequality you want to test against zero.
Choose the x-domain and the sample step for graph detail.
Set decimal places for cleaner output.
Press the calculate button to see the result, graph, detected roots, restrictions, and interval estimate.
Use the CSV button to download sampled rows. Use the PDF button to save the result area as a report.
This calculator helps you study how a function behaves across a chosen x-range. It works with polynomial expressions and rational expressions. You can enter coefficients directly. That makes it easy to test many classroom examples. It also works well for homework checks and quick revisions.
A graph shows more than a final yes or no answer. It reveals where the curve crosses the x-axis. It also shows where the function stays above or below zero. Those regions are the key to solving inequalities. When the graph rises above the axis, the function is positive. When it falls below, the function is negative.
Rational inequalities are different because the denominator can create restrictions. If the denominator becomes zero, the function is undefined there. Those x-values cannot be included in the solution. This page scans for denominator roots and lists them as vertical restrictions. That makes the result easier to read and safer to use.
The calculator samples many x-values across the selected domain. It evaluates the numerator, denominator, and final function value for each point. Then it checks whether the chosen inequality holds. The satisfying rows are grouped into interval estimates. This method is useful for graph-based understanding and numerical review.
The sampled table gives a clear audit trail. You can inspect x-values one by one. That is useful when you want to confirm a sign change or study behavior near a restriction. The CSV export helps with spreadsheet work. The PDF export helps with reports, assignments, and printed notes. Together, the graph, summary, and table provide a practical workflow for inequality analysis.
Yes. Enter numerator coefficients for every case. Enter denominator coefficients only when you want a rational function. If the denominator is all zeros, the page treats the function as a polynomial.
It gives a graph-based numerical estimate on the selected domain. That is excellent for checking work, visual study, and interval inspection. Exact symbolic factoring is not required for the page to work.
Undefined points happen when the denominator becomes zero. Rational functions cannot use those x-values. The calculator excludes them from valid sample rows and lists the related denominator roots.
The sample step controls graph detail and interval precision. Smaller steps create more points and a smoother plot. Larger steps calculate faster but may show less detail near roots or restrictions.
The interval is built from sampled x-values, not from symbolic factorization. That means the result is an estimated interval on your chosen domain. Smaller steps usually improve the estimate.
Yes. Every coefficient field accepts decimal values. That helps when you need to graph scaled models, transformed functions, or non-integer examples from applied math problems.
Choose a domain wide enough to show all expected roots and restrictions. If you already know where the key behavior happens, use a tighter range to get a more detailed graph.
The CSV button exports all sampled rows. The PDF button saves the result section, including the summary and graph. That makes sharing and record keeping much easier.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.