Enter functions, choose values, and inspect composition. See g(x), h(x), and a reusable results table. Export calculations easily for study, checking, and revision later.
The composition rule is:
h(x) = f(g(x))
First evaluate the inner function g(x). Then place that result into the outer function f(x).
Example:
f(x) = x^2 + 1
g(x) = 2x - 3
h(x) = f(g(x)) = (2x - 3)^2 + 1
The calculator follows that same order for a single value and for every row in the generated table.
Example functions: f(x)=x^2+1 and g(x)=2x-3
| x | g(x)=2x-3 | h(x)=(2x-3)^2+1 |
|---|---|---|
| 0 | -3 | 10 |
| 1 | -1 | 2 |
| 2 | 1 | 2 |
| 3 | 3 | 10 |
A function composition calculator combines two rules into one result. Here, the inner function runs first. Then the outer function uses that output. This method builds the composite function h(x)=f(g(x)). It is useful in algebra, precalculus, and calculus. It also helps with substitution practice. Many students understand composition faster when they can see both the symbolic form and the numeric value together.
Composite functions appear in many math topics. They describe layered change. One rule transforms the input. Another rule transforms the new value. This structure appears in graph transformations, inverse functions, trigonometric models, and real applications. It also appears in science and engineering formulas. When you compose functions correctly, you protect the order of operations. That order matters because f(g(x)) is usually different from g(f(x)).
The table gives more than a single answer. It lets you inspect patterns across many x values. You can compare the inner function and the composite output side by side. That makes it easier to spot symmetry, growth, turning points, and unexpected restrictions. It also helps when checking homework steps. Teachers can use the table to explain how one input moves through two connected functions in sequence.
This solver reduces common substitution mistakes. It shows f(x), g(x), and the combined expression h(x). It also evaluates g(x) first, which matches the correct composition order. If you test a range of values, you can verify whether your algebra is consistent. Export tools also make review easier. You can save results, compare attempts, and keep a clean record for assignments, revision, or class discussion.
It means you apply g(x) first. Then you place that output into f(x). The result is one composite function called h(x).
Order changes the result. In most cases, f(g(x)) is not equal to g(f(x)). The inner function must be evaluated before the outer function.
Yes. You can use functions such as sin(x), cos(x), tan(x), log(x), log10(x), sqrt(x), abs(x), and exp(x).
Yes. Use the caret symbol for exponents. For example, write x^2, (x+1)^3, or (2*x-1)^2.
Your expression may have a syntax issue, mismatched parentheses, or a domain problem. Examples include dividing by zero or taking sqrt of a negative value.
The table shows x, g(x), and h(x) across a range. It helps you inspect patterns, verify steps, and compare outputs quickly.
Yes. The calculator includes CSV export for spreadsheet work and PDF export for saving, sharing, or printing your results.
Yes. It helps confirm substitution order, symbolic composition, and final values. It is useful for practice, checking, and quick review.
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.