# MAPPING AND FUNCTIONS

**WEEK NINE**

**SS1 FURTHER MATHS FIRST TERM**

**MAPPING AND FUNCTIONS**

**CONTENT**

- Types of Function
- Application of Function

**SUB TOPIC: TYPES OF FUNCTION**

**Inverse Function:**The inverse of a function is usually written as^{-1}meaning it is a mapping from The inverse^{-1}need not be a function. The only occasion where^{-1}is a function occurs when^{-1}is both and Onto. Suppose on the set into set illustrated in set of ordered pairs and arrow diagrams below.

**-2**

**1**

**3**

**9**

**5**

**-1**

**A**

**B**

*f*

If we consider the relation which reverses i.e. ^{-1} ^{-1} then ^{-1} is a function because it is both one – one and onto. This is so because itself is both One – One and Onto. Hence, a function has an inverse, ^{-1}, if it is both One – One and Onto.

In the illustration above, the rule which reverses the function, (i.e i.e the inverse function can be derived as follows:

Let . Hence, ^{-1}(

Check:

^{-1}(^{-1-1}

**:**The mapping which takes an element Onto itself is called the identity mapping. The mapping shown in the arrow diagram below is an identity mapping.

**a**

**b**

**c**

**c**

**a**

**b**

*f*

It is both One – One and onto. It has a unique property that the domain, the co – domain and the range are equal.

**Constant Mapping:**A constant mapping is a mapping in which all element in the co – domain of are mapping into a single element in the co – domain.

**A**

**g**

**B**

**Circular Functions:**Trigonometric functions are circular function. The range of the function of sine and cosine of an angle is not affected by increasing or decreasing the value of the angle (i.e the domain) and this range revolves around and ; though it is not the case with tangent function. The table below gives the range and domain of three circular functions.

Let |
Domain |
Range |

**Note:**

We cannot consider trigonometrically functions as One – to – One mappings, unless we restrict the domain. Within a domain of {0^{0}, 360^{0}}, any line drawn parallel to the between is bound to cut the sine and cosine curves in more than one place. This evidently shows that the sine and cosine functions in the domain 00 to 3600 are not One – to – One. On the other hand, a One – to –one function can be obtained from trigonometrically function by suitably restricting the domain.

For example:

is not One –to – One,

^{0},^{0}],

**Logarithmic and Exponential Functions:**The exponential function and the logarithmic function_{a}are mutual inverses:^{-1}and^{-1}

The exponential function is expressed by the equation: ^{x} or is the exponential number 2.7182818…

The general exponential function is given by: ^{x}, where and because ^{ln}**^{a}** or

^{log}

_{e}

^{a}, the general exponential function can be written in the form:

^{xln}

**The inverse function of the general exponential function i.e**

^{a}.^{-1}can be derived thus:

Let ^{-1}

From ^{xlna}; ⇒^{x} ; ^{x} or _{a}. _{a}

**CLASS ACTIVITIES: **

- If
^{-1}stating the value of for which^{-1}is not defined. Hence, find^{-1} - State the domain and range of the following:
- b.
^{2}

**SUB TOPIC: APPLICATION OF FUNCTIONS – SOLUTIONS TO PROBLEMS ON FUNCTION.**

- Sketch the graph of where state its domain and range.

**Solution:**

can also be written as:

By this definition, can never be negative. It therefore has the following sketch:

**0**

Its domain is real line and is range is

- If find:
- A function defined by
- The function defined by find the values of for which

**Solution: **

^{2} ^{2}

^{2}

^{ 2}

The values of for which are and

- Given the functions
^{232}. Simplify, as far as possible, the expressions:

**Solution:**

^{232}

^{323232}

^{543243232}.

^{5432}

^{32}^{2}

^{232}

^{32}

^{2}

^{2}

0 0 0

- Two functions are defined by:
^{2}where - Find the largest domain and the corresponding range of each of the two functions.
- Find the inverse function
^{-1}of and hence the composite function^{-1}where^{-1}means^{-1}first, then .

**Solution:**

- The largest domain of , while the range of . The largest domain of , while the range of
- Let
^{-1}Hence,^{-1}

^{-12}

_{22} take this expression to the possible end.

**CLASS ACTIVITIES:**

- A function is defined for the range

Sketch and state the domain and range of

- Let

^{2} , Find the domain and range of find also ^{-1-1}.

**PRACTICE EXERCISE:**

**Objective Test:**

- Two functions are defined on the set of real numbers by and
^{2}find the value of^{-1}a. 12 b. 11 c. d. . - A function is defined on the set of real numbers, by . Find
^{-1}

a). ^{-1 } b). ^{-1 } c). ^{-1 } d). ^{-1 }

**Essay Questions:**

- Given that
^{-12}and - If find the value of for which
^{-1 }is not defined. - Find the image of under the mapping
- The mapping is defined on the set of real numbers. Given that f(0)=-3 and f(1)=2, determine the values of b and c. hence find f(-2) and f(2).
- If

**ASSIGNMENT**

- If for what value of constant p is (fog)=(gof)?
- Two functions f and g are define on the set of R of real numbers by: find f[g(x-1)].
- State the range and the domain of each of the following: (i).
- Let and G(x) = 2x + 1 respectively. Find FoG and GoF. Hence deduce the inverse mapping FoG.
- The functions f and g are defined by determine the composite functions g[f(x)] and f[g(x)].

**KEY WORDS**

**RELATION****MAPPING****FUNCTION****IMAGES****DOMAIN****SUB DOMAIN****ELEMENTS****RANGE****ONE-TO-ONE****ONTO****COMPOSITE****CONSTANT****INVERSE****ARROW DIAGRAM.**