Problem 39 - Thomas Calculus

Question

As in Section \(4.3,\) a function \(f(x)\) increases on an interval \(I\) if for any two points \(x_{1}\) and \(x_{2}\) in \(I,\) $$ x_{2}>x_{1} \Rightarrow f\left(x_{2}\right)>f\left(x_{1}\right) $$ Similarly, a function decreases on \(I\) if for any two points \(x_{1}\) and \(x_{2}\) in \(I,\) $$ x_{2}>x_{1} \Rightarrow f\left(x_{2}\right)

Step-by-Step Solution

Answer

Increasing and decreasing functions are one-to-one because different inputs yield different outputs.

Step 1: Understand Definitions

Recall that a function \( f(x) \) is increasing on an interval \( I \) if whenever \( x_2 > x_1 \), it follows that \( f(x_2) > f(x_1) \). Similarly, it's decreasing if \( x_2 > x_1 \) implies \( f(x_2) < f(x_1) \). We aim to prove that a function that is either increasing or decreasing is one-to-one, meaning different inputs lead to different outputs.

Step 2: Prove for Increasing Functions

Suppose for contradiction that an increasing function is not one-to-one. Then there exist distinct \( x_1 \) and \( x_2 \) (i.e., \( x_1 eq x_2 \)) such that \( f(x_1) = f(x_2) \). Without loss of generality, assume \( x_2 > x_1 \). By definition of increasing, \( x_2 > x_1 \) implies \( f(x_2) > f(x_1) \), which contradicts the assumption that \( f(x_1) = f(x_2) \). Hence, an increasing function must be one-to-one.

Step 3: Prove for Decreasing Functions

Similarly, assume a decreasing function is not one-to-one. There exist \( x_1 \) and \( x_2 \) such that \( x_1 eq x_2 \) and \( f(x_1) = f(x_2) \). Assume \( x_2 > x_1 \). According to the definition of decreasing, \( f(x_2) < f(x_1) \), which contradicts \( f(x_1) = f(x_2) \). Thus, a decreasing function must also be one-to-one.

Key Concepts

Increasing Functions

When we talk about functions, an increasing function is one where, as the input value increases, the output value also rises. You can visualize this as a graph that climbs upwards, from left to right. Here's how it works: given any two points, say \(x_1\) and \(x_2\), within an interval \(I\), if \(x_2\) is greater than \(x_1\), then the function's output at \(x_2\) is higher than at \(x_1\).

This means that for an increasing function, the mathematical expression \(x_2 > x_1 \Rightarrow f(x_2) > f(x_1)\) holds true. Because of this property, increasing functions are inherently one-to-one. That means each input has a unique output. There's no way two different inputs can result in the same output, which supports their one-to-one nature.

Decreasing Functions

Decreasing functions work in the opposite way of increasing functions. In these functions, as the input increases, the output does exactly the opposite—it decreases. Imagine a graph that descends from left to right. For any two points \(x_1\) and \(x_2\), if \(x_2\) is greater than \(x_1\), then the function's output at \(x_2\) will be less than at \(x_1\).

This is represented mathematically as \(x_2 > x_1 \Rightarrow f(x_2) < f(x_1)\). Because this relationship remains consistent across the entire interval, decreasing functions are also one-to-one. Like increasing functions, they don't allow two different inputs to map to the same output, ensuring that each input has a unique associated output.

Contradiction Proof

A contradiction proof is a logical approach often used in mathematics to establish the truth of a proposition. For proving that increasing and decreasing functions are one-to-one, we can use this clever technique. Let's break down how it works:

1. **Assume the opposite is true:** Start by assuming the function is not one-to-one. This means there are distinct inputs, \(x_1\) and \(x_2\) where \(x_1 eq x_2\), but \(f(x_1) = f(x_2)\).

2. **Apply function's property:** Next, apply the properties of increasing/decreasing functions. If the function is increasing, then \(x_2 > x_1\) implies \(f(x_2) > f(x_1)\), and if it's decreasing, \(x_2 > x_1\) implies \(f(x_2) < f(x_1)\).

3. **Reach a contradiction:** The assumption that \(f(x_1) = f(x_2)\) directly contradicts these properties of the function. Since this contradiction arises from the false assumption, it proves that the function must be one-to-one.

Function Properties

Functions come with a variety of properties that define their behavior and make them predictable across their domain. These characteristics include how they increase or decrease, continuity, and periodicity, among others. When we talk specifically about increasing or decreasing functions, a few key traits stand out:

* **Monotonicity:** Increasing and decreasing functions are both examples of monotonic functions, which means they consistently move in one direction across their intervals.
* **Inverses:** Because these functions are one-to-one, each function has an inverse. Inverse functions allow us to reverse the mapping from outputs back to inputs.
* **Continuity:** Often, increasing and decreasing functions are continuous over their domains. This means they don't have breaks, jumps, or gaps and their graphs can be drawn without lifting the pencil.

Understanding these properties allows us to predict and interpret function behavior in mathematical contexts. It helps explain why increasing and decreasing functions are one-to-one, showcasing their reliability and consistency across their domains.