Families of Functions
If mathematicians are cooks, then families of functions are their ingredients. Each family of functions has its own flavor and personality. Before you learn to combine functions to create an infinite number of potential models, you need to get a clear idea of the name of each function family and how it acts.
The identity function is the simplest function and all straight lines are transformations of the identity function family.
The reciprocal function is also known as a hyperbola and a rational function. It has two parts that are disconnected and is not defined at zero. Simple electric circuits are modeled with the reciprocal function.
So far all the functions can be grouped together into an even larger function family called the power function family.
The logarithmic function is closely related to the exponential function family. Many people confuse the graph of the log function with the square root function. Careful analysis will show several important differences. The log function is the basis for the Richter Scale which is how earthquakes are measured.
The sine graph is one of many periodic functions. Periodic refers to the fact that the sine wave repeats a cycle every period of time. Periodic functions are extremely important for modeling tides and other real world phenomena.
The absolute value function is one of the few basic functions that is not totally smooth.
The logistic function is a combination of the exponential function and the reciprocal function. This curve is very powerful because it models population growths where the maximum population is limited by environmental resources.
Which functions always have a positive slope over the entire real line?
Compare and contrast the graphs of the two functions:
Describe the symmetry among the function families discussed in this concept. Consider both reflection symmetry and rotational symmetry.
Some function families have reflective symmetry with themselves:
Some function families are rotationally symmetric:
Some pairs of function families are full or partial reflections of other function families:
For 1-10, sketch a graph of the function from memory.
11. Which function is not defined at 0? Why?
12. Which functions are bounded below but not above?
To see the Review answers, open this PDF file and look for section 1.1.