Evaluate Composite Functions Calculator

Example Data Table

Formula Used

For two functions, the main rule is:

(f ∘ g)(x) = f(g(x))

For the reverse order, use:

(g ∘ f)(x) = g(f(x))

For three functions, evaluate the innermost function first:

(f ∘ g ∘ h)(x) = f(g(h(x)))

Domain Notes

• G: any square root input must be zero or greater.
• A valid starting x can still create an invalid middle value.

Supported operators include +, -, *, /, %, and ^. Supported functions include sin, cos, tan, sqrt, abs, log, ln, exp, floor, ceil, round, sec, csc, and cot.

How to Use This Calculator

1. Enter f(x), g(x), and h(x). You may type only the right side.
2. Choose the composition order, such as f(g(x)) or g(f(x)).
3. Enter the x value where the chain should be evaluated.
4. Select the angle mode when trigonometric expressions are used.
5. Set decimal precision and optional table range values.
6. Press the submit button to show the result above the form.
7. Use the CSV or PDF button to save the result.

Composite Function Evaluation Guide

What the Calculator Does

Composite functions link one rule inside another rule. The inner function is evaluated first. Its output becomes the input for the next function. This calculator supports two function chains and longer three function chains. It also accepts a custom order. You can compare f(g(x)), g(f(x)), f(g(h(x))), and repeated patterns.

Why Order Matters

Order changes the answer in many problems. A square after a shift is not the same as a shift after a square. The tool shows each substitution step. This makes mistakes easier to find. It also helps students see why composition is not usually commutative.

Inputs You Can Test

Enter expressions with x as the variable. Use powers, roots, fractions, and common functions. Examples include x^2+1, sqrt(x+4), sin(x), ln(x), and 1/(x-2). Select radians or degrees when trigonometric functions are used. Choose decimal precision for cleaner results.

Using the Table

The value table tests the same composition across many x values. Set a start value, end value, and step size. The table is useful for graphing points. It also reveals undefined values. If a denominator becomes zero, or a root becomes invalid, the row displays an error note.

Interpreting Results

Read the first line as the full composition. Then review the steps from inside to outside. Each row shows the function used, the input received, and the output produced. The final value is the answer for the selected x. Export the work when you need a record for class, tutoring, or checking homework.

Good Practice

Use parentheses when an expression could be unclear. Write 2*(x+1) instead of 2x+2 when you want a grouped input. Check restrictions before trusting a final answer. Composite functions inherit limits from every function in the chain. A valid starting value can still create an invalid middle value. This is why step by step evaluation is important.

When to Export

Use the CSV file for spreadsheet checks. Use the PDF file for a simple report. Both downloads include the chosen functions, the selected chain, the starting value, and the generated table. This helps keep the original problem beside the answer. It is also helpful when comparing several possible composition orders. Save each version with clear file names.

FAQs