Problem 42 - Thomas Calculus

Question

A magic trick You may have heard of a magic trick that goes like this: Take any number. Add \(5 .\) Double the result. Subtract \(6 .\) Divide by \(2 .\) Subtract \(2 .\) Now tell me your answer, and I'll tell you what you started with. Pick a number and try it. You can see what is going on if you let \(x\) be your original number and follow the steps to make a formula \(f(x)\) for the number you end up with.

Step-by-Step Solution

Answer

The final equation ends with the original number: \(x\). The trick always reveals the starting number.

Step 1: Add 5 to your number

Let's denote the original number as \(x\). The first step in the trick is to add \(5\) to your number. This can be represented as: \(x + 5\).

Step 2: Double the result

Next, you double the result from the first step. So you take the result \(x + 5\), and multiply it by 2. This gives: \[2(x + 5) = 2x + 10\]

Step 3: Subtract 6

The third step is to subtract \(6\) from the result of the doubling. This results in: \[2x + 10 - 6 = 2x + 4\]

Step 4: Divide by 2

Now, divide the result by 2. This simplifies to: \[\frac{2x + 4}{2} = x + 2\]

Step 5: Subtract 2

Finally, subtract \(2\) from the result to find the number you indicate to the magician. \[(x + 2) - 2 = x\]

Step 6: Conclusion: Reverse Engineering the Trick

The final result \(x\) is your original number. This shows that no matter what number you start with, by following the steps, you will always return to your original number, thus revealing the magic trick's secret.

Key Concepts

Functions Illuminate Relationships

In algebra, a function is a relationship between a set of inputs and a set of possible outputs. More simply, it’s like a machine that takes an input, applies a series of operations, and gives back an output. Here, in the magic trick, you start with the input number, let’s call it \(x\), and follow the operations to get a new number, or output. This can be described as a function \(f(x)\).

The operations in the magic trick, like adding, doubling, and so on, define the function. Each step transforms \(x\) until you arrive at your output. The sequence of operations can be thought of as defining a unique path for each input.

• Add \(5\) – transforms number to \(x + 5\)
• Double – transforms to \(2(x + 5)\)
• Subtract \(6\) – results in \(2x + 4\)
• Divide by \(2\) – simplifies to \(x + 2\)
• Subtract \(2\) – returns to \(x\)

Thus, the function returns \(x\), highlighting that functions can reveal neat, often surprising relationships, just like the magic trick.

The Art of Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to make sense of problems and solve them. In this magic trick, each step requires understanding and breaking down the operations in a logical manner. This involves sequential thinking and recognizing that each step depends on the previous one.

Understanding the sequence means knowing why each operation leads closer to the original number:\(x\). This practice can also help with problem-solving in more complex algebraic functions and builds confidence in handling mathematical procedures.

• Recognize patterns – like how every operation connects to the next.
• Breakdown steps – see how each operation affects \(x\).
• Predict outcomes – understand why you always end up with \(x\).

Such reasoning is essential in algebra, enabling one to decode even the most complex of equations, or in this case, a magic trick!

Reverse Operations Demystify the Trick

Reverse operations mean working backward through a series of steps. In the magic trick, to discover how you always arrive back at \(x\), you can imagine reversing the steps. With reverse operations, you can see how easily you could retrace your steps and undo each operation.

Going backward from your answer to see your starting number employs reverse operations, essentially undoing each part of your function \(f(x)\).

• Undo subtract \(2\) – result becomes \(x + 2\)
• Undo divide by \(2\) – turns into \(2x + 4\)
• Undo subtract \(6\) – becomes \(2x + 10\)
• Undo double – returns to \(x + 5\)
• Undo add \(5\) – finally arrives back at \(x\)

These steps illustrate that reverse operations allow you to trace how each correction leads you back to the original number. It’s a powerful method to understand and solve functions because it undoes all operations precisely as they were performed. Thus, understanding reverse operations not only unlocks the magic trick but also gives insight into solving mathematical functions in everyday algebra.