In this lesson, our instructor Vincent Selhorst-Jones introduces the Properties of Functions. Youll learn about increasing and decreasing constants, how to find intervals by looking at the graph, about maximums and minimums. The average rate of change, zeros in a function, roots, and x-intercepts are explored. Lastly, Vincent will go over even/odd functions and graphs. Four fully explained example problems close out the lesson.
Over an interval of x-values, a given function can be increasing, decreasing, or constant. That is, always going up, always going down, or not changing, respectively. This idea is easiest to understand visually,
so look at a graph to find where these things occur.
We talk about increasing, decreasing, and constant in terms of intervals: that is, sections of the horizontal axis. Whenever you talk about one of the above as an interval, you always give it in parentheses.
An (absolute/global) maximum is where a function achieves its highest value. An (absolute/global) minimum is where a function achieves its lowest value.
A relative maximum (or local maximum) is where a function achieves its highest value in some "neighborhood". A relative minimum (or local minimum) is where a function achieves its lowest value in some "neighborhood". [Notice
that these aren't necessarily the highest/lowest locations for the entire function (although they might be), just an extreme location in some interval.]
We can refer to all the maximums and minimums of a function (both absolute and relative) with the word extrema: the extreme values of a function.
We can calculate the average rate of change for a function between two locations x1 and x2 with the formula
f(x2) − f(x1)
It is often very important to know what x values for a function cause it to output 0, that is to say, f(x) = 0. This idea is so important, it goes by many names: the zeros of a function, the roots, the x-intercepts. But these all
mean the same thing: all x such that f(x) = 0.
An even function (totally different from being an even number) is one where
f(−x) = f(x).
In other words, plugging in the negative or positive version of a number gives the same output. Graphically, this means that even functions are symmetric around the y-axis (mirror left-right).
An odd function (totally different from being an odd number) is one where
f(−x) = − f(x).
In other words, plugging in the negative version of a number gives the same thing as the positive number did, but the output has an additional negative sign. Graphically, this means that odd functions are symmetric around the origin (mirror left-right and up-down).
Properties of Functions
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.