Ch. 2.3 Getting Information from the Graph of A Function

This paragraph introduces the topic of extracting information from the graph of a function. The instructor explains the concept of looking at the graphical representation of a function to gather insights, similar to analyzing its algebraic form. The focus is on interpreting points on the graph as 'x f(x)' instead of '(x, y)', emphasizing the function's output given an input. The domain and range of a graph are also discussed, with a method to determine the domain by drawing vertical lines to the x-axis and identifying the x-values that the graph covers.

The second paragraph delves into the specifics of analyzing a graph's domain and range. It explains how to find the domain by drawing vertical lines from the graph to the x-axis and shading the x-values that are covered by the graph. The range is determined by looking at the y-values that the function maps to, without including endpoints unless specified by the graph. Additionally, the paragraph discusses how to compare two functions for equality and inequality by finding their points of intersection and examining the intervals where one function is greater or less than the other.

This paragraph discusses the concepts of local and global maxima and minima within the context of function graphs. It defines a local maximum as the highest output value in a neighborhood of a certain input value, with no other outputs in that vicinity being larger. Similarly, a local minimum is the lowest output with no smaller outputs nearby. The paragraph also explains global maxima and minima as the highest and lowest outputs over the entire domain of the function. The instructor uses visual examples to illustrate these concepts and introduces mathematical notation to describe them.

The fourth paragraph focuses on intervals of increase and decrease for functions represented graphically. It defines an interval of increase as a range where, for any two inputs within that interval, the smaller input value results in a smaller output value than the larger input value. This indicates that the function is rising as x increases. Conversely, an interval of decrease is where the smaller input value results in a larger output value, indicating the function is falling. The paragraph also clarifies that intervals of increase or decrease are always open intervals, not including endpoints, and provides examples to illustrate these concepts.

The final paragraph continues the discussion on intervals of increase and decrease, emphasizing the distinction between strict and non-strict inequalities. It explains that a strict inequality indicates the function is always increasing or decreasing without any flat portions, while a non-strict inequality allows for constant values within the interval. The paragraph also illustrates how to identify these intervals on a graph, using examples of a function that first decreases and then increases, indicating separate intervals of decrease and increase. The summary concludes with a reminder of the importance of open intervals in this context.