Functions

Functions

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Table of Contents

1. Intervals - Slide 3

2. Functions - Slide 4

Key Points - Slide 5

3. Injective functions - Slide 12

​4. Surjective functions - Slide 15

5. Bijective functions - Slide ​16

6. Composite functions - Slide 18

7. Inverse of functions - Slide ​20

1. Intervals

2. Functions

Key points

The range of the function 'f ' is the set of ALL images. If R represents the range of a function, then R is a subset of the codomain, if the codomain is the set of all Real numbers.

Two functions f and g can only be equal if the produce the same output for the same input, i.e. f(x) = g(x) for all values of x.

Domain, codomain and range are written as sets when listing elements.

Range is the set of elements in the codomain that was mapped.

A function can be written as a set of ordered pairs.

f(x) = 2x+1

Domain = { 1,2,3}

Range = {3,5,7}

Codomain = {3,4,5,6,7,8}

The image of 1 is 3. The pre-image of 7 is 3.

Set of ordered pairs

f = { (1,3) , (2,5) , (3,7) }

Note:

Some functions can only be defined as a set of ordered pairs and not by a function.

A function can also be identified for a plotted graph using the vertical line test. The graph represents a function if it is only possible for a vertical line to cut the graph at only one point.

The function f : [ -2, 4 ] --> R is defined by f(x) = 3x + 2. Draw a graph of this function and state its range.

Range [ -4, 14 ]

Note:

Drawing a graph with shaded dots at the end means it is closed and with not shaded dots mean not inclusive. Finding the range is the 1st and last y values / the minimum and maximum y values for the function given its domain.

When no interval is given. (quadratic function)

Method 2

When an interval is given (quadratic function)

3. Injective functions (one- to -one)

A function is said to be injective if each element in the domain is assigned an unique and distinct element in the codomain. In other words, each element in the range has exactly 1 pre-image.

Note:

*A function is said to be many-to-one if there is at least one element in the range that has more than one pre-images.

* An injective function can have an unmapped element in its CODOMAIN but all elements in the DOMAIN must be mapped.

To determine if a function is injective

1. A function f(x) is said to be injective if f (a) = f(b) implies a = b . (Can be used in simple cases to show if a function is injective)

2. The horizontal line test. With the graph of a function, it is determined to be injective if it is only possible for a horizontal line to cut the graph at only one point.

e.x

4. Surjective functions (Onto)

5. Bijective Functions

6. Composite functions

egs

7. Inverse of functions

Inverse of a quadratic

Method 2

Graph of the inverse function

When drawn on the same axes, the graph of y = f(x) and y =f′(x)are reflections of each other in the line y = x . If the graphs intersect, they do so at points which lie on the line y=x. 

Note: 

The same scale for both axes must be used to bring out the symmetry properly about the y=x line. 

E.g

A function f(x) is defined by the set of ordered pairs :

f = { (0, -2) , (1 , -1) , (2, 2) , (3 , 7) , (4, 14) }. It is given that f'(x) exists . Draw the graph of y = f(x) and y = f'(x) on the same axes.