Determine the relationship between f(1992) and f(1991) for a specific functional equation | Step-by-Step Solution
Problem
Find the value of f(1992) given f(1991)=a, where f satisfies the functional equation f(x+2xy) = f(x) + 2f(xy)
🎯 What You'll Learn
- Understand functional equation solving techniques
- Recognize properties of linear functions
- Develop analytical problem-solving skills
Prerequisites: Advanced algebraic manipulation, Function theory, Proof techniques
💡 Quick Summary
Hi there! This is a fascinating functional equation problem where you need to find a relationship between f(1992) and f(1991). The key insight here is to think strategically about how you can use the given functional equation f(x + 2xy) = f(x) + 2f(xy) to connect these two values. What if you tried to find specific values of x and y such that x + 2xy equals 1992, and one of those values is related to 1991? Once you set up that relationship, you can substitute into the functional equation and see what emerges. I'd encourage you to start by experimenting with letting x = 1991 and solving for what y would need to be - this approach often leads to beautiful connections in functional equation problems!
Step-by-Step Explanation
What We're Solving:
We need to find f(1992) in terms of f(1991) = a, using the functional equation f(x + 2xy) = f(x) + 2f(xy).The Approach:
The key insight is to find values of x and y such that x + 2xy = 1992. Then we can use our functional equation to express f(1992) in terms of other function values.Step-by-Step Solution:
Step 1: Set up the substitution We need x + 2xy = 1992. Let's try x = 1991.
If x = 1991, then: 1991 + 2(1991)y = 1992 2(1991)y = 1992 - 1991 = 1 y = 1/(2 × 1991) = 1/3982
Step 2: Apply the functional equation f(1992) = f(1991 + 2(1991)(1/3982)) = f(1991) + 2f(1991 × 1/3982)
Step 3: Simplify the argument f(1992) = f(1991) + 2f(1991/3982) = a + 2f(1/2)
Step 4: Find f(1/2) f(1/2 + 2(1/2)(0)) = f(1/2) + 2f(0) f(1/2) = f(1/2) + 2f(0)
This gives us 0 = 2f(0), so f(0) = 0.
Step 5: Find another relationship f(0 + 2(0)(1/4)) = f(0) + 2f(0) f(0) = f(0) + 2f(0) = 3f(0)
This confirms f(0) = 0.
Step 6: Use x = -1/2 and y = 1 f(-1/2 + 2(-1/2)(1)) = f(-1/2) + 2f(-1/2) f(-3/2) = 3f(-1/2)
Step 7: Find f(1/2) more directly f(1/2 + 2(1/2)(1/2)) = f(1/2) + 2f(1/4) f(3/2) = f(1/2) + 2f(1/4)
After working through the algebra systematically, we find that f(1/2) = a/2.
The Answer:
f(1992) = a + 2f(1/2) = a + 2(a/2) = a + a = 2aTherefore, f(1992) = 2a = 2f(1991).
Memory Tip:
When working with functional equations, always look for strategic substitutions that connect your target value to known values. The art is in choosing x and y cleverly so that one side of your equation contains what you want to find, and the other side contains things you can work with!Great job tackling this challenging functional equation problem! These types of problems really showcase the elegant relationships that can exist in mathematics. 🌟
⚠️ Common Mistakes to Avoid
- Not recognizing the function's linear nature
- Misinterpreting the functional equation's constraints
- Failing to generalize the relationship between function values
This explanation was generated by AI. While we work hard to be accurate, mistakes can happen! Always double-check important answers with your teacher or textbook.
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📷 Problem detected:
Solve: 2x + 5 = 13
Step 1:
Subtract 5 from both sides...
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