Properties of functions (§3.3)

Earlier this week, you looked at properties of graphs in the plane, which you can now think of 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 now!

Symmetry: even and odd functions

As a graph can be symmetric with respect to an axis or the origin, so a function can be even or odd.

A function is even if:
- its graph is symmetric with respect to the y-axis;
- f(−x) = f(x) always.
A function is odd if:
- its graph is symmetric with respect to the origin;
- f(−x) = −f(x) always.

A function's graph is hardly ever symmetric with respect to the x-axis; the only way that this can happen is when the output of the function is always 0.

Intercepts: roots (or zeroes) and initial values

To find the x-intercepts, you must solve the equation

f(x) = 0.

The solutions to this equation are the roots of f. The book calls these the zeroes of f, but I find that term confusing. There are many shortcuts for this for certain kinds of functions, some of which we'll use later on.

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

(0, f(0)).

This f(0) is the initial value of f. There are sometimes shortcuts for this, but they're hardly worth the bother; 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 y-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, 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, from the Latin for ‘touching’, whose slopes are instantaneous rates of change.)

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, and even though the same behaviour is also true on any smaller interval.) Mostly you will just look for these on the graph, rather than calculating rates of change to find them; the calculations, while possible using only Algebra, become much easier using Calculus.

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. (Unfortunately, different people sometimes use slightly different definitions of these terms, which can be clarified by adding extra adjectives. So the local extrema here are really strict two-sided local extrema. If you take another course, look at another website, or read another book, then you may be expected to use a weaker meaning.)

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; you're unlikely to ever be expected to use them differently.)

Note the grammar used with this terminology (both local and absolute): when a function f has an extremum at b, the extremum is f(b). Calculating extrema usually requires 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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