Calculator
Current Formula Used
Outer function: f(x) = x² + 4
Inner function: g(x) = 2x
Composition rule: f(g(x)) = a(mx+n)² + b(mx+n) + c
Expanded result: f(g(x)) = 4x² + 4
With the default settings, g(x)=2x and f(x)=x²+4. So f(g(x)) becomes f(2x). Replace x in f(x) with 2x. The result is (2x)²+4, which simplifies to 4x²+4.
Example Data Table
| x | g(x) | f(g(x)) |
|---|---|---|
| -2 | -4 | 20 |
| -1 | -2 | 8 |
| 0 | 0 | 4 |
| 1 | 2 | 8 |
| 2 | 4 | 20 |
How to Use This Calculator
- Keep the default values to solve g(x)=2x and f(x)=x²+4.
- Change a, b, and c to build another quadratic outer function.
- Change m and n to build another linear inner function.
- Enter the x value where you want to evaluate f(g(x)).
- Set table start, end, and step values for comparison rows.
- Press Calculate to view the result above the form.
- Use CSV or PDF buttons to save the output.
About Function Composition
What This Tool Solves
Function composition means placing one function inside another function. In this calculator, the inner function is g(x). The outer function is f(x). The final expression is written as f(g(x)). This means g(x) is evaluated first. Its result then becomes the input for f. The default problem uses g(x)=2x and f(x)=x²+4. After substitution, f(g(x)) becomes f(2x). The expression is then expanded and simplified.
Why Composition Matters
Composition appears in algebra, graphs, modeling, and applied problems. It helps describe chained actions. One rule changes the input first. Another rule changes the new result. For example, a machine may double a number first. Then another machine may square it and add four. The combined rule gives one direct expression. This saves time when many values must be tested.
Advanced Options
This calculator lets you edit every coefficient. The outer rule can be any quadratic expression. The inner rule can be any linear expression. The tool expands the composition automatically. It also evaluates the expression at your chosen x value. The derivative value shows the local rate of change. The vertex helps describe the turning point. The discriminant helps review possible roots. These extra outputs make the page useful for deeper study.
Checking Your Work
Use the table to compare several x values. This is helpful when drawing a graph. It is also useful when checking homework answers. The CSV file works well for spreadsheets. The PDF file is useful for notes and sharing. Always review the substitution step first. Most mistakes happen when replacing x inside the outer function. Write each step clearly. Then simplify powers, products, and constants in order.
Frequently Asked Questions
1. What is f(g(x))?
f(g(x)) means the output of g(x) is placed inside f(x). You solve g(x) first. Then use that result as the input for f.
2. What is the default problem?
The default problem uses g(x)=2x and f(x)=x²+4. The calculator finds f(g(x)), expands it, and evaluates it at your chosen x value.
3. What is f(g(x)) for the default values?
For g(x)=2x and f(x)=x²+4, replace x in f(x) with 2x. The result is (2x)²+4, which simplifies to 4x²+4.
4. Can I change the functions?
Yes. You can change a, b, and c for the outer quadratic. You can also change m and n for the inner linear function.
5. Why is there a table?
The table shows several x values with their matching g(x) and f(g(x)) results. It helps you compare values and prepare graph points.
6. What does the derivative output mean?
The derivative output gives the rate of change of the composed expression at your selected x value. It is useful for advanced algebra review.
7. What does the vertex mean?
The vertex is the turning point of the expanded quadratic composition. It shows where the curve reaches a minimum or maximum when applicable.
8. Do the download buttons save my current result?
Yes. The CSV and PDF buttons use the current form values. They save the formula, selected x value, result, and related details.