Properties of functions (§3.3)

Last week, we looked at properties of graphs in the plane, which we can now think of as properties of relations. Since we can think of a function as a special kind of relation, these properties also apply to functions. However, since functions had a very 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.

Intercepts: roots and zeroes

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, and they're often very helpful.

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

(0,f(0)).

There is no special name for f(0). There are sometimes shortcuts for this, but they're hardly worth the bother.

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 not defined, 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) 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.

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

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.

Mostly you will just look for these on the graph, rather than calculating rates of change.

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 is anything that's either a local maximum or a local minimum. Warning: When a function f has a local extremum at b, the local extremum is f(b).

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 is anything that's either an absolute maximum or an absolute minimum. Calculating these usually requires calculus, so mostly you'll just look for these on the graph too.

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