Calculator
Formula Used
The composite function is written as:
h(x) = g(f(x))
For the requested value, the calculator uses:
h(3) = g(f(3))
First, the calculator substitutes 3 into f(x). Then it sends that answer into g(x). The final answer is h(3).
How to Use This Calculator
- Enter the inner function in the f(x) box.
- Enter the outer function in the g(x) box.
- Keep the target x value as 3 for h(3).
- Select precision and angle mode if needed.
- Set the table range for nearby values.
- Press the calculate button.
- Use the CSV or PDF buttons to save the result.
Example Data Table
| f(x) | g(x) | f(3) | h(3) = g(f(3)) |
|---|---|---|---|
| 2*x+1 | x^2-4 | 7 | 45 |
| sqrt(x+6) | 3*x+2 | 3 | 11 |
| x^2-5 | abs(x)+1 | 4 | 5 |
Nested Function Work Made Clear
A nested function can look small, yet it hides two jobs. First, the inner rule f receives the input. Then the outside rule g receives the answer from f. This calculator follows that order every time. It is designed for students, tutors, worksheet writers, and anyone checking algebra steps. The target value is three by default, because the question asks for h(3). You may still change the input when you want to test another point.
Why Order Matters
Composition is not the same as simple multiplication. The notation g(f(x)) means that f acts first. Its output becomes the new input for g. If f(3) equals seven, the next step is g(7), not g(3). Many mistakes happen when users skip that middle value. Showing f(3) beside h(3) makes the process easier to audit.
Advanced Input Support
The form accepts arithmetic signs, powers, parentheses, constants, and common functions. You can write rules such as 2*x+1, x^2-4, sqrt(x+6), or sin(x). The parser checks the expression before calculation. It also prevents division by zero and invalid operators. Precision control helps when results contain decimals. Angle mode helps when trigonometric rules are used.
Practical Uses
This tool is useful for homework, quick examples, lesson pages, and answer keys. It can compare several nearby x values in a generated table. The table helps users see how the composite function changes around three. Exports also make the result easier to save. A CSV file works well for spreadsheets. A PDF works well for printing or sharing.
Reading the Result
The result panel appears above the form after submission. It shows the entered rules, f(3), and the final h(3). It also displays the substitution chain. This layout keeps the answer near the header and easy to find. Use the example table when you need a model. Then replace the sample rules with your own expressions.
Validation Notes
Use the asterisk for multiplication, because 2x is not accepted. Keep parentheses balanced. Start with simple rules, then add powers or functions. When a value is outside a function domain, the calculator reports an error. This helps users fix inputs before copying the final answer. Clear errors protect the result and guide faster worksheet corrections today.
FAQs
What does h(x)=g(f(x)) mean?
It means f works first. Its answer becomes the input for g. The final output is called h(x).
How do I calculate h(3)?
Put 3 into f(x). Then put that result into g(x). The calculator shows both steps and the final answer.
Can I change 3 to another value?
Yes. The target input field is editable. Keep it as 3 when your question specifically asks for h(3).
Which multiplication symbol should I use?
Use the asterisk symbol. Write 2*x instead of 2x. This keeps the expression clear and easier to validate.
Does the calculator support powers?
Yes. Use the caret symbol for powers. For example, x^2 means x squared, and x^3 means x cubed.
Can I use square roots?
Yes. Use sqrt(x) for square roots. The calculator will show an error when the square root input is negative.
Why is f(3) shown separately?
It is the middle value. Showing it helps verify the substitution before the calculator applies g to that output.
What do the export buttons save?
The CSV and PDF buttons save the entered rules, the target value, the main result, and the generated table.