Lecture 2: Domain and range of functions - Trinity College Dublin

Lecture 2: Domain and range of functions

Victoria LEBED, lebed@maths.tcd.ie MA1S11A: Calculus with Applications for Scientists

October 3, 2017

Yesterday we saw examples of functions defined on certain parts of the real line R only--e.g., the square root x x, defined for x 0; defined by di erent formulas on di erent parts of R--e.g., the absolute value x |x|: |x| = x, x 0, -x, x < 0.

These parts are typically intervals, rays, and various combinations thereof.

Today we will

learn how to handle these intervals, rays, etc. mathematically (this will be our tools), discuss how to determine for which values of the independent variable a function is defined, and what values it takes (this is where we will apply our tools).

1 Reminder: intervals

[a, b] stands for the closed interval consisting of all x for which a x b. Mathematically, this can be wri en in a very e icient way:

[a, b] = { x R | a x b }. Read: [a, b] is the set of all real values x such that a x b holds. A more "human" reading: [a, b] is the set of all real x between a and b.

Similarly, the open interval (a, b) is defined by (a, b) = { x R | a < x < b }.

You might meet an alternative notation ]a, b[ for the open interval (a, b).

Half-closed intervals: [a, b) = { x R | a x < b }, (a, b] = { x R | a < x b }.

2 Reminder: rays

An open end of an interval may assume infinite values, in which case an interval becomes a ray: for example,

[a, +) = { x R | a x }, is closed ray, while

(-, b) = { x R | x < b },

is an open ray.

(-, +) is a symbol notation for the whole real line R.

Intervals and rays are the simplest parts of R. More complicated parts are obtained from these "building blocks" using operations: intersection, union, di erence.

3 Operations on intervals: example

Alice and Bob are flatmates. Yesterday Alice was at home from 6am to midday, and Bob from 1am to 7am.

estion 1. When were both of them at home? Solution. Consider the intervals A = [6, 12], and B = [1, 7]. The question asks to compute the intersection of these time intervals:

A B = [6, 12] [1, 7] = [6, 7]. So, Alice and Bob were both at home from 6am to 7am.

estion 2. When was at least one of them at home? Solution. We need to compute the union of our time intervals:

A B = [6, 12] [1, 7] = [1, 12].

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download