Composite Function Game

Composite Function Game
This game will require you to perform algebraic composition, that is plugging the expression of g(x) into the expression of f(x), and then simplifying the result into a standard polynomial. Scroll down for a detailed explanation.
 

 

How to Play the Composite Function Game
The Strategy:
Identify the inner function (g(x)).
Take that entire expression and substitute it everywhere you see an x in the outer function (f(x)).
Expand the parentheses and combine like terms.

Example 1 (Linear):
f(x) = 2x + 3
g(x) = 3x - 1(f ∘ g)(x) = 2(3x - 1) + 3
Simplify: 6x - 2 + 3 = 6x + 1

Example 2 (Quadratic):
f(x) = x² + 1g(x) = x - 2
(f ∘ g)(x) = (x - 2)² + 1
Simplify: (x² - 4x + 4) + 1 = x² - 4x + 5

Formatting Tips:
Use ^2 or ² for squares (e.g., x^2 + 4x + 4).
Ensure you combine your constants into a single number.

What is Composite Function
In mathematics, a Function Composition (also known as Function of a Function) is an operation where the output of one function becomes the input for another.

The Mechanics: How it Works
When you see the notation (f ∘ g)(x), it is read as “f of g of x.” It can also be written as:
f(g(x))
In this setup, there is an inner function and an outer function:
Inner Function g(x): This is solved first.
Outer Function f(x): This takes the result of g(x) and processes it.

Two Ways to Solve It

1. Numerical Evaluation (Finding a Number)
   If you are given a specific value for x, you solve it from the inside out.
   Example: Let f(x) = x + 5 and g(x) = 2x. Find f(g(3)).
   Step 1: Solve the inner function: g(3) = 2(3) = 6.
   Step 2: Plug that result into the outer function: f(6) = 6 + 5 = 11.
   Final Result: 11.
   

2. Algebraic Composition (Finding a New Rule)
   Sometimes you don’t have a number; you want a single formula that represents both steps combined.
   Example: Let f(x) = x² and g(x) = x - 3. Find f(g(x)).
   Step 1: Look at f(x). Everywhere you see an x, leave a blank space: ()².
   Step 2: Stuff the entire expression of g(x) into that blank: (x - 3)².
   Step 3: Simplify: x² - 6x + 9.
   

Order Matters.
Composition is not commutative. This means the order in which you apply the functions usually changes the result.

This video gives a clear, step-by-step approach to composite functions.

• Show Video

 

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