Composite Functions

TL;DR

• This is a function composition test: given g(x)=f^{-1}(x) and several mapping points, reverse-engineer f's mappings, then compute f(f(f(6))).
• Good for testing: whether the model handles inverse mapping correctly, and whether it makes errors during multiple compositions.
• Production tip: require the model to output a mapping table and show intermediate values after each composition step.

Background

This prompt tests an LLM's mathematical capabilities by prompting it to evaluate a function composition.

How to Apply

Follow the "build the mapping first, then compute step by step" approach:

1. From g(x)=f^{-1}(x) we get: f(g(x)) = x (i.e., reverse g's pairs to get f's pairs)
2. List f's mapping in a table (e.g., f(5)=0, f(7)=4, etc.)
3. Compute sequentially: f(6) → f(f(6)) → f(f(f(6)))

How to Iterate

1. Force output of a mapping table (structured) before giving the final answer
2. Add self-check: cross-verify f and g (check whether g(f(x)) = x holds at the given points)
3. Adversarial testing: add more points, introduce distractor points, or make f non-injective to see if the model catches contradictions

Self-check Rubric

• Did it correctly understand the inverse relationship and construct f's mapping?
• Is the composition computed step by step with correct intermediate values?
• Did it check whether the mapping is consistent (no conflicts)?

Practice

Exercise: change the target to f(f(2)), f(f(f(9))), etc., and require the model to output:

• mapping table
• intermediate value at each step
• brief consistency check

Prompt

Suppose g(x) = f^{-1}(x), g(0) = 5, g(4) = 7, g(3) = 2, g(7) = 9, g(9) = 6.
What is f(f(f(6)))?

Code / API

OpenAI (Python)

from openai import OpenAI

client = OpenAI()

response = client.chat.completions.create(
    model="gpt-4",
    messages=[
        {
            "role": "user",
            "content": "Suppose g(x) = f^{-1}(x), g(0) = 5, g(4) = 7, g(3) = 2, g(7) = 9, g(9) = 6. What is f(f(f(6)))?",
        }
    ],
    temperature=1,
    max_tokens=256,
    top_p=1,
    frequency_penalty=0,
    presence_penalty=0,
)

Fireworks (Python)

import fireworks.client

fireworks.client.api_key = "<FIREWORKS_API_KEY>"

completion = fireworks.client.ChatCompletion.create(
    model="accounts/fireworks/models/mixtral-8x7b-instruct",
    messages=[
        {
            "role": "user",
            "content": "Suppose g(x) = f^{-1}(x), g(0) = 5, g(4) = 7, g(3) = 2, g(7) = 9, g(9) = 6. What is f(f(f(6)))?",
        }
    ],
    stop=["<|im_start|>", "<|im_end|>", "<|endoftext|>"],
    stream=True,
    n=1,
    top_p=1,
    top_k=40,
    presence_penalty=0,
    frequency_penalty=0,
    prompt_truncate_len=1024,
    context_length_exceeded_behavior="truncate",
    temperature=0.9,
    max_tokens=4000,
)

Reference

• Sparks of Artificial General Intelligence: Early experiments with GPT-4 (13 April 2023)