Question

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=2 x-3 $$

Short answer

The function \(f(x) = 2x - 3\) has no critical numbers and it is increasing for all \(x\), over the interval \(-\infty < x < +\infty\).

Step-by-step solution

Differentiate the Function

The derivative of the function \(f(x) = 2x - 3\) is \(f'(x) = 2\).

Find Critical Numbers

Critical numbers are found by setting the derivative equal to zero and solving for \(x\). However, for this function \(f'(x) = 2\) is a constant and doesn't equal to zero at any point. So, the function does not have any critical numbers.

Determine the Intervals of Increase or Decrease

As the derivative \(f'(x) = 2\) is positive for all \(x\), this means that the function \(f(x) = 2x - 3\) is increasing for the interval \(-\infty < x < +\infty\).

Key concepts

Derivative of a Function

Understanding the derivative of a function is central to calculus and plays a pivotal role in identifying the behavior of graphs. In its simplest form, a derivative represents the rate at which a function is changing at any given point. It is the slope of the tangent line to the function's curve at that point. For a linear function such as \(f(x) = 2x - 3\), the derivative is a constant, \(f'(x) = 2\). This constant value indicates that the function has a steady rate of change and the slope is the same across its entire domain.

In practical terms, if we were to graph the linear function, we could draw a straight line with a slope of 2. This line would consistently rise, moving two units up for every one unit it moves horizontally. The derivative helps predict this linear behavior and is an essential tool for analyzing more complex functions where rates of change vary across different sectors of the graph.

Intervals of Increase and Decrease

When exploring the nature of functions, it's useful to determine where they are increasing or decreasing. To do this, we can utilize the derivative of the function. If the derivative is positive over an interval, the function is increasing on that interval. Conversely, if the derivative is negative, the function is decreasing. This relationship between derivatives and the growth or decline of a function is instrumental in sketching the behavior of graphs over certain ranges.

For instance, the function \(f(x) = 2x - 3\) has a derivative of \(f'(x) = 2\), a positive number, suggesting that the function is increasing everywhere. Hence, there are no intervals of decrease and the entire real number line is an interval of increase for the function. This constant rate of increase is typical for linear functions and is a straightforward example of how derivatives can quickly inform us about the nature of a function's graph.

Graphing Utility

Graphing utilities are invaluable tools for visualizing mathematical functions and confirming analytical findings. They allow us to plot functions, observe their behaviors, and understand their properties without extensive manual calculations. In educational settings, they provide a visual reinforcement of concepts such as slopes, intercepts, and curvature.

For a linear function like \(f(x) = 2x - 3\), graphing it would illustrate a straight line that intersects the y-axis at -3 (the y-intercept) and has a slope of 2 (as indicated by the derivative \(f'(x)\)). The positive slope confirms our earlier analysis that the function is increasing over its entire domain. Using a graphing utility, students can see the impact of this positive slope directly on the graph, reinforcing their understanding of derivatives, slope, and the behavior of linear functions.