Question

In Problems \(1-10\), use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ f(x)=3 x+3 $$

Short answer

The function is increasing for all \( x \in \mathbb{R} \).

Step-by-step solution

Recognize the Function

The function given is a linear function of the form \( f(x) = 3x + 3 \). Linear functions have a constant slope, meaning they are either always increasing, always decreasing, or constant throughout their domain.

Determine the Slope

For a linear function \( f(x) = mx + b \), the slope \( m \) determines the function's monotonicity. Here, the slope \( m \) is 3.

Apply the Monotonicity Theorem

The Monotonicity Theorem tells us that if the slope of a linear function is positive (\( m > 0 \)), then the function is increasing, while if the slope is negative (\( m < 0 \)), the function is decreasing. Since \( m = 3 > 0 \), the function is increasing.

Conclude the Monotonicity

Since the function has a positive slope across its entire domain, \( f(x) = 3x + 3 \) is increasing for all \( x \in \mathbb{R} \). There are no intervals where it is decreasing.

Key concepts

Understanding Linear Functions

Linear functions are one of the simplest types of functions and are foundational in mathematics. They are of the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This type of function graphs as a straight line. The function given in the exercise, \( f(x) = 3x + 3 \), fits this form perfectly.
Here are some basic characteristics of linear functions:

• The graph of a linear function is a straight line.
• The line crosses the y-axis at point \( b \), known as the y-intercept.
• The slope \( m \) determines the angle and direction of the line.

Linear functions consistently increase, decrease, or remain constant, which makes them predictable and straightforward to work with.

What is Slope?

Slope is a crucial concept in understanding linear functions. It defines the steepness and direction of a line. Represented by \( m \) in the equation \( f(x) = mx + b \), the slope tells us how much \( y \) changes for a unit change in \( x \).

Consider the following details about slope:

• A slope of zero means the line is flat, or horizontal.
• A positive slope indicates the function is increasing as \( x \) increases.
• A negative slope shows the function is decreasing as \( x \) increases.

In the exercise, the slope is \( 3 \), positive, meaning the function consistently rises as \( x \) moves from left to right.

Understanding Increasing Functions

An increasing function is one that moves upwards as it progresses along the x-axis. This means that as \( x \) values grow larger, the \( y \) values of the function also grow higher. For linear functions with a positive slope, like \( f(x) = 3x + 3 \), this is always true.
Key features of increasing functions include:

• For any \( x_1 < x_2 \), \( f(x_1) < f(x_2) \).
• The graph of the function continuously rises.
• It reflects continual growth or increase over its domain.

The Monotonicity Theorem helps us conclude if the function is increasing or decreasing by merely evaluating its slope. Here, with a slope of \( 3 \), the function is confirmed to be increasing across its entire domain.

Approaching Calculus Problems

Solving calculus problems often involves understanding deeper mathematical concepts. However, in the context of linear functions and their monotonicity, the calculus is simplified.
Some essential strategies include:

• Identify the type of function you are working with and recognize its standard forms.
• Use the function's slope to apply the Monotonicity Theorem when assessing if it increases or decreases.
• Remember that linear functions maintain a constant slope, which simplifies the analysis to determining if the slope is positive, negative, or zero.

For the exercise in question, the calculus problem boiled down to evaluating the slope, which led us to determine that the function is increasing throughout its domain. No deep calculus techniques were needed here!