Problem 34 - Calculus: One and Several Variables

Question

Give the domain of the function and sketch the graph. $$f(x)=2 x+1$$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = 2x + 1\) is \(-\infty < x < +\infty \). The function has a slope of 2, a y-intercept of 1, and its graph is a straight line that crosses the y-axis at \(y = 1\).

Step 1: Define The Domain of the Function

With the linear function \(f(x) = 2x + 1\), the function is defined for all real numbers. Therefore, the domain of the function \(f(x) = 2x + 1\) is \(-\infty < x < +\infty \).

Step 2: Identify the Slope and Y-intercept

In the equation \(f(x) = 2x + 1\), the coefficient of \(x\), which is 2, represents the slope of the line, and the constant term, which is 1, represents the y-intercept of the graph. Therefore, the line has a slope of 2 and intercepts the y-axis at \(y = 1\).

Step 3: Sketch the Graph

First, plot the y-intercept on the y-axis at \(y = 1\). Then, use the slope to find a second point. Each unit rise is met with a unit run of the slope, which is 2 in this case. Therefore, you will move up 2 units and to the right 1 unit from the y-intercept to determine the second point. Draw the line through these two points to get the graph.

Key Concepts

Domain of a function

The domain of a function refers to the set of all possible input values (usually represented by the variable \( x \)) for which the function is defined. For linear functions like \( f(x) = 2x + 1 \), the domain is as vast as possible. This means that you can input any real number into the function without any worry of it becoming undefined or problematic. In mathematical terms, the domain of \( f(x) = 2x + 1 \) is represented as

• \(-\infty < x < +\infty \)

For linear equations, unless otherwise restricted, the domain will always be all real numbers. This trait makes them incredibly predictable and easy to work with in various scenarios.

Slope

The slope of a line is a measure of its steepness. It tells us how much the line rises or falls as we move along it. In our function,

• \( f(x) = 2x + 1 \)

The slope is represented by the coefficient of \( x \), which is 2. This means for every increase of 1 unit in \( x \), the function value increases by 2 units. It's often denoted as \( m \) in the slope-intercept form of a line \( y = mx + b \). The slope gives insights into the direction of the line:

• A positive slope, like 2, means the line ascends as it moves from left to right.
• A negative slope would imply the opposite.

This linear relationship is important, especially when predicting future points on the line.

Y-intercept

The y-intercept is the point at which the line crosses the y-axis. It indicates the value of the function when \( x \) is zero. In the function

• \( f(x) = 2x + 1 \)

The y-intercept can be found by looking at the constant term, which is 1. This means the graph of the function will cross the y-axis at the point

• \( (0, 1) \)

Understanding the y-intercept allows us to start the graph sketching process from a known point and is crucial for translating a linear equation into a visual representation. It's a fixed starting point for sketching the graph.

Graph sketching

Graph sketching involves drawing the visual representation of a mathematical function on a coordinate plane. For the function \( f(x) = 2x + 1 \), starting the sketch is straightforward when you know the slope and y-intercept.To begin:

• First, locate the y-intercept on the graph, which is the point \((0, 1)\).
• From this point, use the slope. With a slope of 2, move up 2 units and to the right 1 unit to find your next point \( (1, 3) \).
• Draw a straight line through these points.

The line should extend infinitely in both directions, illustrating the concept of a continuous function with no breaks or gaps. This visual process helps solidify understanding of the function's behavior and provides a tangible way to interpret linear equations.