Composite Function Game (Evaluate)

Composite Function Game (Evaluate)
This game will require you to find the answer to g(x) first, then plug that answer into f(x). Scroll down for a detailed explanation.
 

 

How to Play the Composite Function Game (Evaluate)
Step 1: Solve the “Inner” Function (g)
Take the Initial Input (x) provided in the box and plug it into the g(x) (Emerald) function.
Example: If g(x) = 2x and x = 3, your intermediate result is 6.
Step 2: Solve the “Outer” Function (f)
Take the result you just got from Step 1 and plug it into the f(x) (Indigo) function. Do not use the original x here.
Example: If f(x) = x + 5, plug in your 6. The final answer is 6 + 5 = 11.

A Quick Walkthrough
If the screen shows:f(x) = x² + 1
g(x) = x + 3
Initial Input: x = 2
Work g(2): 2 + 3 = 5
Work f(5): 5² + 1 = 25 + 1 = 26
Enter: 26 into the answer box and hit Submit.

Common Pitfalls
Wrong Order: Do not plug x into f first. In the notation (f ∘ g), the function closest to the x (on the right) is always the first step.
The “Double x” Mistake: Make sure you don’t use the original x for both functions. The second function must use the output of the first one.

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.
   

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

• Show Video

 

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