Properties of functions (§2.3)

You can think of properties of graphs in the plane as properties of relations. Since you can think of a function as a special kind of relation, these properties also apply to functions. However, since functions had a different historical development, all of the names are different!

Symmetry: even and odd functions

As a graph can be symmetric with respect to one of the axes or to the origin, so a function can be even or odd.

A function f is an even function if:
- its graph is symmetric with respect to the vertical axis, meaning that (−a, b) is on the graph whenever (a, b) is;
- f(−x) = f(x) whenever f(x) exists.
A function f is an odd function if:
- its graph is symmetric with respect to the origin, meaning that (−a, −b) is on the graph whenever (a, b) is;
- f(−x) = −f(x) whenever f(x) exists.
A function f is a zero function if:
- its graph is symmetric with respect to the horizontal axis, meaning that (a, −b) is on the graph whenever (a, b) is;
- f(x) = 0 whenever f(x) exists.

(We mostly don't care about zero functions; there is exactly one zero function with any given domain D, defined by f(x) = 0 for x ∈ D.)

Intercepts: roots (or zeroes) and initial values

To find the horizontal intercepts, you must solve the equation

f(x) = 0.

The solutions to this equation are the roots of f; these are also called the zeroes of f (but I find that term confusing). That is, if r is a root, then (r, 0) is a horizontal intercept of the graph. There are many shortcuts for this for certain kinds of functions, some of which we'll use later on. Note that there may be any number of roots, from zero to infinitely many.

The graph of a function can only have one vertical intercept, which is easy to compute:

(0, f(0)).

This f(0) is the initial value of f. If f(0) doesn't exist, then the graph has no vertical intercept. There are sometimes shortcuts for this, but they're usually not worth the bother; instead, you just plug in 0 and evaluate.

Slopes: rates of change

A linear function has the form

f(x) = mx + b.

Its graph is a line with slope m and vertical intercept (0, b). When talking about the function, we call this number m the rate of change of f. (The slope of a vertical line is undefined, but that's OK, since a vertical line is not the graph of a function.)

Few functions are linear. However, given two inputs a and b of the function (and assuming that the function is defined between a and b), we can imagine a line drawn through the corresponding points on the graph, which is called a secant line, (so called from the Latin for ‘cutting’) and calculate its slope:

m = [f(b) − f(a)] ÷ [b − a].

This number (the slope of that secant line) is the average rate of change of f from a to b. (If you take Calculus, then you'll learn how to find tangent lines, so called from the Latin for ‘touching’, whose slopes are instantaneous rates of change. By the way, the secant and tangent from Trigonometry are not related to these lines.)

For a particular function on a particular nontrivial interval, sometimes the average rate of change of that function between any two points in that interval always has the same sign (positive or negative).

The function is increasing on the interval if the average rate of change is always positive.
The function is decreasing on the interval if the average rate of change is always negative.
The function is constant on the interval if the average rate of change is always zero.

(The textbook always expects you to give the largest nontrivial open intervals on which a function is increasing, decreasing, or constant, even if the behaviour continues to one or both endpoints, but it's worth giving a closed or half-closed interval if possible.) Mostly you will just look for these on the graph, rather than calculating rates of change to find them; the calculations, while usually possible using only Algebra, become much easier using Calculus. (To be very precise, these are strictly increasing and strictly decreasing functions; a function is weakly increasing if the average rate of change is always positive or zero, and similarly for weakly decreasing functions.)

A function has a local maximum at an input b if it is increasing on an interval [a, b] and decreasing on an interval [b, c]. It has a local minimum at b if it is decreasing on an interval [a, b] and increasing on an interval [b, c]. A local extremum (also called a relative extremum) is anything that's either a local maximum or a local minimum. (To be very precise, these are really strict two-sided local extrema. If you allow the functions to be weakly increasing or decreasing to the sides, then you get weak local extrema; if you allow them to be undefined on one side, then you get one-sided local extrema.)

A function has an absolute maximum at an input b if f(a) ≤ f(b) for any a in the domain of f. It has an absolute minimum at b if f(a) ≥ f(b) for any a in the domain of f. An absolute extremum (also called a global extremum) is anything that's either an absolute maximum or an absolute minimum. (This is the weakest and default meaning of these terms, although sometimes people talk about strict or two-sided absolute extrema.)

Note the grammar used with this terminology (both local and absolute): when a function f has an extremum at c, the extremum is f(c). Calculating extrema is much easier with Calculus, so mostly you'll just look for them on the graph in this course. (The traditional plurals of all of these ‘‑um’ words end in ‘‑a’; again, this comes from Latin.)

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