Check a function on one semi-open interval
Use x, pi, e, +, -, *, /, ^, parentheses, and supported single-input functions.
Formula used
For [a, b)
If x is smaller than a or equal to b, the restricted result is undefined.
For (a, b]
If x is equal to a or greater than b, the restricted result is undefined.
The calculator first tests interval membership. It evaluates the expression only when the entered input is accepted.
How to use this calculator
- Enter a function using x as the input variable.
- Enter the lower boundary a and upper boundary b.
- Select [a, b) or (a, b] to set endpoint inclusion.
- Enter the x value that you want to test.
- Choose radians or degrees for trigonometric functions.
- Set sample points and decimal places when needed.
- Select Calculate restriction to view the result above the form.
- Export the generated table as CSV or save a printed PDF copy.
Understanding semi-open interval restrictions
A function restriction narrows the inputs accepted by a rule. A semi-open interval keeps one endpoint and excludes the other. The notation [a, b) includes a but excludes b. The notation (a, b] excludes a but includes b. This distinction changes whether boundary values are valid. It also changes the final function value.
Restricted functions appear in calculus, programming, modelling, and classroom exercises. A formula may work for many numbers. A task may permit a smaller interval. The restriction controls the usable domain. It does not change the underlying expression. Instead, it determines which inputs receive an accepted output. A calculator should report both the domain decision and the computed value.
Start by writing the expression with x as its variable. Simple examples include x^2, 3*x-4, and sqrt(x+5). Choose lower and upper boundaries. The lower boundary must be smaller than the upper boundary. Then choose the bracket pattern. Square brackets mean included. Parentheses mean excluded. A semi-open interval uses one of each. This tool checks rules before evaluating the expression.
For [a, b), a value equals a is allowed. A value equals b is rejected. Interior values are allowed. For (a, b], the reverse endpoint rule applies. A value equals a is rejected. A value equals b is allowed. Values outside either boundary always fail. The calculator displays it clearly. That helps prevent small notation mistakes from becoming incorrect solutions.
The expression is evaluated only after the x value passes the interval test. This order matters. An expression might produce a number for an excluded endpoint. That number is not an accepted restricted-function result. Division by zero, invalid square roots, and undefined logarithms need attention. The calculator stops and explains such issues. It uses safe supported operators, functions, constants, and the x variable.
Sample values make the restriction easier to inspect. The generated sample table places points inside the interval. It avoids the excluded endpoint. This is useful when graphing or checking trends. Use more sample points for a broad interval. Use fewer points for a quick review. The table can be exported as a CSV file. It can also be saved through the browser print dialog.
Precision controls the number of displayed decimal places. It does not alter the internal comparison rule. Keep several decimals when boundaries are close together. Use fewer decimals for simple instructional examples. Trigonometric expressions can use degrees or radians. Select the unit matching your formula. Functions such as sin, cos, and tan depend on that choice. Inverse trigonometric outputs follow the same selected angle unit.
A restricted function result has two parts. First, state whether x belongs to the chosen interval. Second, provide f(x) when the input is valid. Show the endpoint notation with the answer. This makes the result easy to audit. Always reread brackets and parentheses before submission.
Frequently asked questions
1. What is a semi-open interval?
It is an interval that includes one endpoint and excludes the other. Examples are [a, b) and (a, b].
2. Which endpoint does [a, b) include?
It includes the lower endpoint a. It excludes the upper endpoint b.
3. Which endpoint does (a, b] include?
It excludes the lower endpoint a. It includes the upper endpoint b.
4. Does the tool calculate f(x) for an excluded input?
No. The restricted result is reported as undefined, even when the unrestricted expression could produce a number.
5. Can lower and upper boundaries be negative?
Yes. Any finite numeric boundaries are accepted when the lower boundary is smaller than the upper boundary.
6. Which functions can I enter?
You can use arithmetic, powers, parentheses, x, pi, e, and the listed supported single-input mathematical functions.
7. Can I use another variable besides x?
No. The calculator uses x as the single input variable. Replace other variable symbols with x before calculating.
8. Do trigonometric functions use degrees?
They use radians by default. Select Degrees before calculating when your expression requires degree-based trigonometry.
9. Why does the calculator show undefined?
The input may be excluded, or the expression may be mathematically undefined. Examples include division by zero and a negative square root.
10. What are the sample points for?
They show valid inputs distributed across the selected interval. Use them to inspect values, check trends, or prepare a graph.
11. Can I download my result table?
Yes. Calculate first, then use Download CSV. You can also use Print / Save PDF from the browser dialog.