Increasing and Decreasing Functions


Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.


y = 1/|x|

Verified step by step guidance

1. First, understand the function y = 1/|x|. The absolute value function |x| affects the graph by making all x-values positive, which means the function is defined for all x except x = 0.
2. Consider the symmetry of the function. Since |x| is symmetric about the y-axis, the function y = 1/|x| is also symmetric about the y-axis. This means the graph will look the same on both sides of the y-axis.
3. Next, analyze the behavior of the function as x approaches 0 from the left and right. As x approaches 0, |x| becomes very small, making 1/|x| very large. Therefore, the function has a vertical asymptote at x = 0.
4. Determine the intervals of increase and decrease. For x > 0, as x increases, |x| increases, making 1/|x| decrease. Thus, the function is decreasing on the interval (0, ∞). For x < 0, as x decreases (approaches 0 from the left), |x| decreases, making 1/|x| increase. Thus, the function is increasing on the interval (-∞, 0).
5. Finally, graph the function. Plot points for various values of x to see the behavior of the function. Remember the symmetry about the y-axis and the vertical asymptote at x = 0. The graph will show the function decreasing on (0, ∞) and increasing on (-∞, 0), with symmetry about the y-axis.

Key Concepts

Increasing and Decreasing Functions

A function is considered increasing on an interval if, for any two points within that interval, the function's value at the second point is greater than at the first. Conversely, a function is decreasing on an interval if the function's value at the second point is less than at the first. Identifying these intervals is crucial for understanding the behavior of the function and can be determined by analyzing the first derivative.

Graphing Functions

Graphing a function involves plotting its values on a coordinate plane, which visually represents its behavior. For the function y = 1/|x|, the graph will show how the function behaves as x approaches zero and as x moves away from zero. Understanding the graph helps in identifying symmetries and the intervals of increase and decrease.

Symmetry in Functions

Symmetry in functions refers to the property where a function exhibits a mirror-like behavior about a specific axis. For example, a function is even if f(x) = f(-x) for all x, indicating symmetry about the y-axis. In the case of y = 1/|x|, the graph is symmetric about the y-axis, which can simplify the analysis of its increasing and decreasing intervals.