Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=3 x-2 $$

Step-by-Step Solution

Answer

The function is increasing everywhere with domain and range \((-\infty, \infty)\). Sketch is a line through points (0, -2) and (1, 1).

Step 1: Identify the Function Type

The function given is \( f(x) = 3x - 2 \). This is a linear function because it is in the form \( y = mx + b \), where \( m = 3 \) (the slope) and \( b = -2 \) (the y-intercept).

Step 2: Sketch the Graph

To sketch the graph of the function \( f(x) = 3x - 2 \), plot the y-intercept \((0, -2)\) and use the slope \( m = 3 \) to find another point. From \( (0, -2) \), move up 3 units and right 1 unit to get to the point \((1, 1)\). Draw a straight line through these points, extending in both directions.

Step 3: Find the Domain

The function \( f(x) = 3x - 2 \) is defined for all real numbers. Thus, the domain \( D \) is \( (-\infty, \infty) \).

Step 4: Find the Range

Since \( f(x) = 3x - 2 \) is a linear function with a non-zero slope, the output can be any real number as well. Therefore, the range \( R \) is \( (-\infty, \infty) \).

Step 5: Determine Intervals of Increase, Decrease, or Constancy

For a linear function with a positive slope, \( f(x) \) is increasing on its entire domain. Since the slope is \( 3 \), which is positive, \( f(x) \) is increasing on \( (-\infty, \infty) \). It does not decrease or remain constant on any interval.

Key Concepts

Understanding Domain and Range of Linear Functions

When dealing with linear functions, knowing how to find the domain and range is essential. A linear function, like our example function \( f(x) = 3x - 2 \), is a straight line on the graph. Generally, for linear functions without any restrictions (like division by zero or square roots of negative numbers in the domain), both the domain and range are all real numbers. This is because a line continues infinitely in both directions and can take any x-value and produce any y-value.

• The domain of \( f(x) \) refers to all possible x-values. In this case, it's \( (-\infty, \infty) \).
• The range refers to all possible y-values or outputs. For linear functions with any real \( m \) (as long as \( m eq 0 \)), like our slope of 3, the range is also \( (-\infty, \infty) \).

Keep in mind that every linear function, unless restricted, will behave the same regarding domain and range. It's quite straightforward!

Sketching the Graph of Linear Functions

Graph sketching begins by identifying key components of the function. With linear functions, these are the slope and the y-intercept. Let's see how that's applied in our given equation, \( f(x) = 3x - 2 \).
First, identify the y-intercept (the point where the line crosses the y-axis). In this case, it's \(-2\), meaning the point \((0, -2)\) on the graph.
Next, use the slope. The slope tells you how steep the line is. Here, a slope of 3 means for every 1 unit you move to the right, you move up 3 units. Starting at \((0, -2)\), move to the point \((1, 1)\), and draw a straight line through these points.

• Y-intercept: where the line crosses the y-axis, point \((0, -2)\).
• Slope = 3: for every 1 unit in x-move, move 3 units up on the y-axis.
• Draw a straight line through these points, extending in all directions.

Remember: Linear graphs are always straight lines!

Function Intervals in Linear Functions

Linear functions are either always increasing, decreasing, or constant, depending on the slope. For \( f(x) = 3x - 2 \), the slope is 3, which is positive. This indicates that the function is always increasing across its domain.
Here's how to determine intervals of behavior:

• Increasing: Occurs when the slope \( m \) is positive, as in this case. Therefore, \( f(x) \) is increasing on \( (-\infty, \infty) \).
• Decreasing: Occurs when the slope \( m \) is negative.
• Constant: Occurs when the slope \( m \) equals zero, leading to a horizontal line.

Since the slope is positive in our linear function, the function continues to rise as x increases, without any intervals of decrease or constancy. Understanding the slope tells us all we need to know about the interval behavior of linear functions.