Common Properties of Functions

Symmetry

a function $f$ is called

• symmetric about $a$ if for all $x$ it holds that
  • $f(a+x)=f(a-x)$
  • moving it about x- or y-axis still preserves symmetry, just $a$ has to adjust
• even if $f$ is symmetric about 0, i.e.
  • $f(x)=f(-x)$
  • moving it along the y-axis still holds the even property
  • examples: $x^2,x^4+6,x^{16}+x^8$
• odd if for all $x$ it holds that
  • $f(-x)=-f(x)$
  • moving it along the y-axis destroys the odd propetry (adding a constant)
  • examples: $x^3,x^1,x^{17}$
  • if $0$ is in the domain, $f$ needs to include the point (0,0)
    • careful, this is not a general definition, $x^2$ still has (0,0), but is even

Monotonicity

increasing if $a<b\Rightarrow f(a)\le f(b)$ decreasing if $a<b\Rightarrow f(a)\ge f(b)$

strictly increasing if $a<b\Rightarrow f(a)<f(b)$ strictly decreasing if $a<b\Rightarrow f(a)>f(b)$

Also applicable to intervals of functions. i.e. $x^2$ is strictly increasing on $[0,\infty ]$

Any constant function, is increasing and decreasing, neither of which strictly.

A local extremum is a point where a function changes its monotonicity

Trigonometric functions are not monotone (they increase and decrease). They are periodical functions because they repeat their values within 2 bounds.

Convexity & Concavity

conVex if opens upwards (smiley face) concAve if opens downwards (frowny face)

when connecting 2 lines on a function the line is either always above (convex) or always below (concave) the function. This must hold through the entire domain (x-range). Functions (like sin, cos) are both convex and concave, but in different domains. Should there be an inflection point within the domain, neither convexity nor concavity can be attributed.

linear function are both convex and concave.

An inflection point is is a point where a graph changes convexity.

there is also strictly convex and strictly concave functions.