1. What is Function Composition?

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

Notation: (f ◦ g)(x) = f(g(x)) means "f composed with g of x".

2. How Does Function Composition Work?

The composition is created by:

Where:

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

Process: First apply g to x, then apply f to the result of g(x).

3. Importance of Function Composition

Applications: Used in mathematics, computer science, physics, and engineering to combine operations or transformations.

4. Using the Calculator

Tips:

• Enter f(x) and g(x) as mathematical expressions using 'x' as the variable
• Examples: "2*x+3", "x^2", "sin(x)", "sqrt(x)"
• Optionally enter an x value to evaluate the composition at that point

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between (f ◦ g)(x) and (g ◦ f)(x)?
A: The order matters! (f ◦ g)(x) = f(g(x)) while (g ◦ f)(x) = g(f(x)) - these are usually different.

Q2: Can I compose more than two functions?
A: Yes! For example, (f ◦ g ◦ h)(x) = f(g(h(x))). You would need to chain multiple compositions.

Q3: What operations are supported?
A: Basic arithmetic (+,-,*,/), exponents (^), and common functions if implemented.

Q4: Why is my composition result not evaluating?
A: Check your function syntax and ensure you're using valid mathematical expressions.

Q5: Can I use variables other than x?
A: This calculator currently only supports 'x' as the variable name.