Section Summary

What is a function? As we have seen, it's a pairing of the values of one variable with values of another variable in such a way that each value of the first variable is paired with exactly one value of the second variable. We express that idea in terms of

• inputs and outputs,
• first column and second column,
• x-coordinates and y-coordinates,
• domain values and corresponding function values,

and in many other ways.

The act of pairing is a process, and it is often important to view functions dynamically, that is, as processes doing something to the input values to produce the output values. On the other hand, a function (that is, the collection of all the paired values) is an object, and thus we can do things to functions as well. In particular, functions can be treated as algebraic objects, albeit more complicated ones than the familiar numbers, literal constants, and variables. We can combine functions by the algebraic operations of addition, subtraction, multiplication, and division. We can form new functions from old ones by taking negatives, reciprocals, and square roots.

In this section we have also studied the important operation of inverting a function, that is, of acting on the function as object to reverse the function as process. As we will do with many topics in this course, we have viewed inversion of functions three ways: symbolically, numerically, and graphically. Each of the three ways is important, and we review them here.

Symbolic inversion  If a function is defined by a formula y = f ( x ), then inversion means interchanging the symbols x and y to get a new equation, x = f ( y ), which defines the inverse relation. If we can solve this equation for y, and if each x determines a unique y, then the resulting equation, y = g ( x ), defines the inverse function. That is, g = f - 1. However, even if we can't solve the new equation for y, there still may be an inverse function — we just won't know a formula for it.

Numerical inversion  This means at least two different — but closely related — things. In the car-loan and calculator-square-root examples, we did numerical calculations to find single values of x when we knew the values of y. That's often easy — with the help of a computer or calculator. And when it's not easy, it's almost always possible — often with the help of a computer. On the other hand, a function may be known only as a table of data. In this case, inversion is trivial — just interchange the columns in the table.

Graphical inversion  This is almost as easy as inverting a data table: Just flip the graph over the line y = x. When you draw graphs with a computer, you see just how easy this is.

Contents for Chapter 1