Properties of Functions

Understanding properties lets you sketch graphs, solve inequalities, and invert functions with confidence. Monotonicity comes first: on an interval $I$, $f$ is strictly increasing when $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$ and strictly decreasing when the inequality reverses, which you can often read straight off the curve or, once the function is differentiable, off the sign of $f'$ — positive means increasing, negative means decreasing. Boundedness answers a different question: $f$ is bounded on $I$ if $\exists M$ with $|f(x)| \le M$ for all $x \in I$, so $\sin x$ is bounded by $1$ on $\mathbb{R}$ while $x^2$ is bounded by $1$ on $[-1,1]$. Extrema mark the turning points — a local maximum at $x_0$ is a point with a neighborhood where $f(x) \le f(x_0)$, a local minimum one where $f(x) \ge f(x_0)$ — and the usual derivative test finds them: $f'(x_0) = 0$ with the sign of $f'$ switching from positive to negative gives a maximum, the reverse switch gives a minimum. Parity is a symmetry statement, and it only makes sense on a domain symmetric about the origin: $f$ is even when $f(-x) = f(x)$, making the graph symmetric about the $y$-axis as $x^2$ and $\cos x$ are, and odd when $f(-x) = -f(x)$, making it symmetric about the origin as $x^3$ and $\sin x$ are. Periodicity repeats the picture: $f$ is periodic if $\exists T \neq 0$ with $f(x+T) = f(x)$ for all $x$, and the smallest positive such $T$ is the fundamental period — $2\pi$ for $\sin x$, $\pi$ for $\tan x$. Finally, convexity tells you which way the curve bends: on $I$, $f$ is convex (concave up) when the chord between any two points on the graph lies above the curve, which analytically means $f''(x) \ge 0$, and concave (concave down) when $f''(x) \le 0$; the points where the curvature changes sign are the inflection points.

Exercises

Exercise 1

Determine where $f(x) = x^3 - 3x + 1$ is increasing or decreasing.

Exercise 2

Is $f(x) = \ln\!\left(\dfrac{1+x}{1-x}\right)$ odd or even on $(-1,1)$?

Summary

Symbols Used in This Article