Problem 11 - Calculus: One and Several Variables

Question

Determine the domain of the function and sketch the graph. $$f(x)=3 x-\frac{1}{2}$$.

Step-by-Step Solution

Answer

The domain of the function \(f(x)=3x-\frac{1}{2}\) is all real numbers. The graph is a rising line from left to right crossing y-axis at -\frac{1}{2}.

Step 1: Finding the domain

For the given function \(f(x)=3x-\frac{1}{2}\), there are no restrictions on x. So, x can be any number among the real numbers. Therefore, the domain is all real numbers.

Step 2: Identify slope and intercept

The given function can be written in the form of \(y=mx+b\), where m is the slope and b is the y-intercept. Here, slope m=3 and y-intercept b=\(-\frac{1}{2}\). The slope is positive, which means the line rises and the intercept is negative, which means the line crosses the y-axis below the origin.

Step 3: Sketch the graph

Start by plotting the y-intercept, which is at (0,-\frac{1}{2}). From the y-intercept, move up 3 units (due to positive slope) and to the right by 1 unit (since we are considering the slope as rise/run). Continue this process to get a left-to-right rising line.

Key Concepts

Real Numbers

When dealing with the domain of functions in mathematics, it's essential to understand the concept of real numbers. Real numbers include all kinds of numbers that can exist along the number line.
They can be whole numbers, fractions, decimals, positive, negative, and even zero.

For the function \( f(x) = 3x - \frac{1}{2} \), the domain is all real numbers because:

• The function is a linear equation.
• There are no divisions by zero, square roots of negative numbers, or any other operations that might restrict the value of \( x \).

In simpler terms, you can plug any real number into this function, and it will give you a valid output.

Linear Function

Linear functions are one of the simplest types of functions you will encounter in algebra. They form a straight line when graphed.
The general form of a linear function is \( y = mx + b \), where:

• \( m \) represents the slope.
• \( b \) is the y-intercept.

In the function \( f(x) = 3x - \frac{1}{2} \), the equation clearly follows the linear function format. This tells us:

• The graph of this function will be a straight line.
• Its properties make calculations and predictions straightforward using the equation's components.

Slope-Intercept Form

The slope-intercept form, \( y = mx + b \), is a way to quickly identify key components of a linear function:

• \( m \) is the slope, which describes the line's steepness and the direction it moves. For our function, \( m = 3 \), so the line moves upwards as it progresses from left to right.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis. Our function has a y-intercept of \(-\frac{1}{2}\), meaning it touches the y-axis at that point.

Using the slope and intercept, we can quickly gain a sense of the entire line's behavior without extensive calculations.

Graph Sketching

Graph sketching is an important skill that helps visually represent functions. It simplifies understanding and analysis of mathematical situations.

To sketch the graph of \( f(x) = 3x - \frac{1}{2} \), follow these steps:

• Start at the y-intercept \((0, -\frac{1}{2})\). Plot this point on your graph.
• Use the slope \( m = 3 \). This means for every 1 unit you move right, you move 3 units up. Mark the next point at \((1, \frac{2}{2})\), which is \((1, 1.5)\).
• Continue plotting points using the same method, connecting them with a straight line.
• Your line should extend infinitely in both directions, reflecting the function's domain: all real numbers.

This approach provides a concrete visual of what the line looks like, aiding in better comprehension.