1. What is Function Composition (f ◦ g)(x)?

Definition: Function composition combines two functions where the output of one function (g) becomes the input of another function (f).

Purpose: It helps in creating complex functions from simpler ones and is fundamental in advanced mathematics and computer science.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( f \) — The outer function
• \( g \) — The inner function
• \( x \) — The input value (optional)

Explanation: The calculator substitutes g(x) into f(x) to create the composition (f ◦ g)(x). If an x value is provided, it calculates the numeric result.

3. Importance of Function Composition

Details: Function composition is essential in calculus, computer programming, and mathematical modeling. It allows building complex operations from simpler components.

4. Using the Calculator

Tips:

• Enter f(x) and g(x) as mathematical expressions (e.g., "x^2 + 3" or "sin(x)")
• Optionally provide an x value to compute a specific result
• Use standard mathematical notation and operators

5. Frequently Asked Questions (FAQ)

Q1: What operations are supported?
A: Basic arithmetic (+, -, *, /), exponents (^), and common functions (sin, cos, tan, log, etc.).

Q2: Can I use multiple variables?
A: No, this calculator only works with single-variable functions (x).

Q3: Why is my calculation showing an error?
A: Check for syntax errors in your functions or undefined operations (like division by zero).

Q4: Does order matter in function composition?
A: Yes! (f ◦ g)(x) is different from (g ◦ f)(x) in most cases.

Q5: Can I compose more than two functions?
A: This calculator handles two functions, but you can chain results to compose more.