Functions

Question 1

Set

Definition

Answer 1

a collection of objects

Question 2

The Natural Numbers

Answer 2

counting numbers from one to infinity increasing by one each time

Question 3

Function

Definition

Answer 3

Let A and B be two sets. A function is a relationship/rule describing a unique element of set B to each element of A

Question 4

Function

Notation

Answer 4

Let A and B be two sets, for a function f,

f : A -> B and f(a) is the element in B assigned to aεA

Question 5

Range/Image

Definition

Answer 5

For a function f : A -> B, the range/image of f is {f(a) : aεA}. It is denoted by f(A). The set f(A) must be contained within B

Question 6

Surjective

Definition

Answer 6

A function f : A -> B is surjective if for all bεB, there exists some object aεA such that f(a) = b

Question 7

Injective

Definition

Answer 7

If each element of the range is only reached from a single element of the domain, the function is injective
A function is injective if f(a1) = f(a2) implies that a1 = a2
In general, any strictly increasing or decreasing function is injective

Question 8

Bijective

Definition

Answer 8

If a function is both injective and surjective then it is bijective
This means that it is possible to uniquely pair up all members of the domain and codomain

Question 9

When is it possible to find the inverse of a function?

Answer 9

You can only find the inverse of a function if it is injective

Question 10

How to find the inverse function?

Answer 10

For a function in terms of x, set the function equal to y so that you have y in terms of x
Rearrange for x in terms of y

Question 11

Composite Functions

Answer 11

The composition of functions f and g written f º g is defined:
(f º g)(x) = f(g(x))

Question 12

Injective Functions - Graph Test

Answer 12

Graph the function

The function is injective if a horizontal line drawn through the graph at any point only crosses the graph once

Question 13

Even Function

Definition

Answer 13

A function is even if,

f(x) = f(-x) for all x

Question 14

Odd Function

Definition

Answer 14

A function is odd if,

f(-x) = -f(x) for all x

Question 15

Domain of sinr

Answer 15

[-π/2, π/2]

Question 16

Range of sinr

Answer 16

[-1, 1]

Question 17

Domain of cosr

Answer 17

[0, π]

Question 18

Range of cosr

Answer 18

[-1, 1]

Question 19

Domain of arcsin

Answer 19

[1, -1]

Question 20

Domain of arccos

Answer 20

[-1, 1]

Question 21

Range of arcsin

Answer 21

[-π/2, π/2]

Question 22

Range of arccos

Answer 22

[0, π]

Question 23

Domain of tanr

Answer 23

(-π/2, π/2)

Question 24

Range of tanr

Answer 24

real numbers

Question 25

Domain of arctan

Answer 25

real numbers

Question 26

Range of arctan

Answer 26

(-π/2, π/2)

Question 27

Left Limit

Definition

Answer 27

Given a function f and a point x0

The left limit is the value that f approaches as x approaches x0 from below

Question 28

Right Limit

Definition

Answer 28

Given a function f and a point x0

The right limit is the value that f approaches as x approaches x0 from above

Question 29

Continuous Function

Definition

Answer 29

If both a left and right limit exist at x0 and they are equal then the function is continuous at x0
If a function is continuous at every point in its domain then it is a continuous function

Question 30

Union

Definition

Answer 30

∪, the union of two sets is a new set that contains all of the elements that are in at least one of the two sets

Question 31

Intersection

Definition

Answer 31

∩, the intersection of two sets is a new set that contains all of the elements that are in both of the two sets