Review of functions

Functions

Elements : input(independent variable), output(dependent variable),

Sets : domain(set of inputs), range(set of outputs)

Rule for assigning each input to exactly one output : relation from domain to range

 

※ Expression of sets with an infinite number of elements

1. Set-builder notation : {x l 1<x<5}

2. Interval notation : (1, 5) = {x l 1<x<5}

 

※ Piecewise-defined functions

: functions defined using different equations for different parts of their domain

(example) f(x) = 3x+1 (x>=2), x^2 (x<2)

 

Representing functions

1. By a table : table of values, it is hard to get a clear picture of the behavior of the function

2. By a graph : a visual picture of the function(plots)

3. By a algebraic formula 

 

※ Vertical line test (to determine whether a set of plotted points represents the graph of a function )

Given a function f, every vertical line that may be drawn intersects the graph of f no more than once.

If any vertical line intersects a set of points more than once, the set of points does not represent a function.

 

A function f is increasing on the interval I if for all x1, x2 ∈ I,

f (x1) ≤ f(x2) when x1 < x2

We say f is strictly increasing on the interval I if for all x1, x2 ∈ I, 

f (x1) < f(x2) when x1 < x2.

We say that a function f is decreasing on the interval I if for all x1, x2 ∈ I,

f (x1) ≥ f(x2) if x1 < x2.

We say that a function f is strictly decreasing on the interval I if for all x1, x2 ∈ I,

f (x1) > f(x2) if x1 < x2.

 

Combining functions

: create a new function by composing two functions

 

Symmetry of functions

If f(x) = f(−x) for all x in the domain of f, then f is an even function. An even function is symmetric about the y-axis.

If f(−x) = −f(x) for all x in the domain of f, then f is an odd function. An odd function is symmetric about the origin.