1. What is Function Composition (F of G)?

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

Notation: \((f \circ g)(x) = f(g(x))\) means "f of g of x".

2. How Does Function Composition Work?

The composition process:

1. Evaluate the inner function g at x
2. Use the result from g(x) as the input for function f

3. Examples of Function Composition

Example 1: If f(x) = x² and g(x) = x + 1
Then (f ∘ g)(x) = f(g(x)) = f(x + 1) = (x + 1)²

Example 2: If f(x) = √x and g(x) = 2x - 3
Then (f ∘ g)(x) = f(2x - 3) = √(2x - 3)

4. Using the Calculator

Instructions:

• Enter f(x) in the first field (e.g., "x^2" or "sqrt(x)")
• Enter g(x) in the second field (e.g., "x+1" or "2x-3")
• Optionally enter an x value to evaluate the composition at that point

5. Frequently Asked Questions (FAQ)

Q1: Is function composition the same as multiplication?
A: No, composition is different from multiplication. f(g(x)) is not the same as f(x)*g(x).

Q2: Can I compose more than two functions?
A: Yes, like (f ∘ g ∘ h)(x) = f(g(h(x))). The calculation proceeds from right to left.

Q3: Is function composition commutative?
A: No, f(g(x)) is generally not equal to g(f(x)). Order matters in composition.

Q4: What's the domain of a composite function?
A: The domain consists of all x in g's domain where g(x) is in f's domain.

Q5: How is this used in real applications?
A: Composition models complex relationships where one process's output feeds into another.