Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Describe the increasing and decreasing behavior of the function. Find the point or points where the behavior of the function changes. \(f(x)=2 x\)

Step-by-Step Solution

Answer

The function \(f(x)=2x\) is increasing for all \(x\) and there are no points where the behavior of the function changes.

Step 1: Analyze the function slope

Identify the function's type. Noting that \(f(x)=2x\) is a linear function, its slope can be obtained directly from the function's equation. The slope is the coefficient of \(x\), which is \(2\).

Step 2: Determine increasing or decreasing behavior

If the slope is positive, the function is increasing; if it's negative, the function is decreasing. Since in our case the slope is \(2\), which is positive, the function is increasing.

Step 3: Find the point where the behavior changes

For linear functions, there are no points of change in behavior as the function increases or decreases constantly. Hence in this case, there are no points where the behavior of function changes.

Key Concepts

Understanding Slope

In the realm of linear functions, the term "slope" signifies a fundamental concept. The slope in a linear equation determines how steep the line is. Imagine a hill - a gentle slope indicates a less steep hill, while a steep slope means a much steeper hill.

For a linear function written in the form of \( f(x) = mx + b \), the slope \( m \) is the number before the \( x \). It tells us how rapidly and in which direction the function changes as \( x \) increases.

• If \( m > 0 \), the function is rising.
• If \( m < 0 \), the function is falling.
• And if \( m = 0 \), the line is flat and doesn't rise or fall.

In our case, the function \( f(x) = 2x \) has a slope of \( 2 \). This positive slope indicates that for every unit increase in \( x \), \( f(x) \) increases by \( 2 \). Thus, the line rises as \( x \) increases.

Behavior of Increasing and Decreasing Functions

Understanding whether a function is increasing or decreasing helps us grasp its general trend. This behavior is tightly linked to the slope.

**Functions can be classified as follows:**

• Increasing Functions: As \( x \) increases, \( f(x) \) also rises. Typically, these functions have a positive slope, as observed in \( f(x) = 2x \).
• Decreasing Functions: When \( x \) increases, \( f(x) \) tends to decrease; a negative slope is typical in such patterns.

With linear functions like \( f(x) = 2x \), the behavior remains consistent — increasing or decreasing constantly. This is why there isn't any change in behavior for linear functions.

An In-depth Look at Function Behavior Analysis

Function behavior analysis is the process of understanding how a function behaves as its input values change. For linear functions, this is straightforward due to their consistent behavior.

**Key Aspects of Function Behavior Analysis:**

• **Slope Analysis:** As seen, identifying the slope helps determine if the function is increasing or decreasing.
• **Zeroes and Intercepts:** Although not directly related to our function's increasing nature, intercepts can give information about where a function crosses axes.
• **Behavior Changes:** In linear functions, these changes don't occur due to the function's uniform rate of change. For other function types, identifying critical points can be challenging but necessary for understanding behavior shifts.

Linear functions, like \( f(x) = 2x \), showcase simplicity in their predictable rise or fall. Unlike quadratic or more complex functions, linear equations don't have curves or turns; they maintain their trend without sudden shifts. This predictability is why linear functions are often some of the first that students learn to understand function behavior.