Functions

Summary

  • A function is defined as a rule which assigns each element of set X, one and only one element of set Y.  i.e set X\quad =\quad \left\{ { x }_{ 1 },\quad x_{ 2 },\quad { x }_{ 3 } \right\}   and set  Y\quad =\quad \left\{ { y }_{ 1 },\quad { y }_{ 2 },\quad { y }_{ 3 } \right\}
  • Domain: elements of set X
  • Range: elements of set Y
  • One to One Function: for each element of set Y, there is a unique corresponding element in set X.
  • Many to One Function: for any element of set Y, there is more than one element in set X.
  • Inverse Function:  f^{ -1 }\left( y \right) \quad =\quad x
  • Composite Function: combine two functions to get a new function f\left[ g\left( x \right) \right]
  • Modulus Function:  f\left( x \right) \quad =\quad \left| x \right|

Let’s suppose we have two sets of numbers:

X\quad =\quad \left\{ { 10,\quad 20,\quad 30,\quad 40 } \right\}
Y\quad =\quad \left\{ { 32,\quad 62,\quad 92,\quad 122 } \right\}

To define a relationship between these two sets we write a function:

f\left( x \right)\quad =\quad 3x\quad +\quad 2 or can be written as f\quad :\quad x\quad \rightarrow \quad 2x\quad +\quad 3

Now we can see that for every value of set X, we have a value for set Y i.e

When x\quad =\quad 10        f\left( x \right)\quad =\quad 3(10)\quad +\quad 2\quad =\quad 32

When x\quad =\quad 20        f\left( x \right) \quad =\quad 3(20)\quad +\quad 2\quad =\quad 62   and so on…

If the following two conditions are satisfied than the relation in called a “Function”:

i) we can say that every element of X has an element in set Y.
ii) every element of X has only one unique element in set Y.

Therefore, a function is defined as a rule which assigns each element of set X, one and only one element of set Y.

Domain and range

Going back to the above example which consisted of two sets:

X\quad =\quad \left\{ { 10,\quad 20,\quad 30,\quad 40 } \right\}
Y\quad =\quad \left\{ { 32,\quad 62,\quad 92,\quad 122 } \right\}

Here elements of set X are known as “Domain” and elements of set Y are known as “Range”.

Hence we can write the domain and range as:

Domain of f = {10, 20, 30, 40}
Range of f = {32, 62, 92, 122}

Note: we cannot define a function if its domain is not mentioned, moreover range of a function may or may not consist of all elements of the set Y.

Example #1

Q. Let  f\quad :\quad x\quad \rightarrow \quad { (x\quad -\quad 4) }^{ 2 }\quad +\quad 8  with domain x > 0.

Find the range of f.

Solution:

Since a function consists of a square, and we know that result of a square is always positive, therefore all values of f(x) will be greater than 8.

f\quad :\quad x\quad \rightarrow \quad { (positive\quad value) }\quad +\quad 8

Hence the range of f is  > 8

One to One Function

If we have a function f such that for each element of set Y (the range), there is a unique corresponding element in set X (the domain), then f is called a one to one function i.e:

given  f\left( x \right)\quad =\quad 2x\quad +\quad 5 from X to Y where X = {4, 7, 8} is one to one as shown below:

Many to One Function

If we have a function f such that for any element of set Y (the range), there is more than one element in set X (the domain), then f is called a many to one function, i.e:

f\quad :\quad x\quad \rightarrow \quad 2{ x }^{ 2 }\quad +\quad 4x\quad -6 is a many to one function as the result x = 1 & -3 both correspond to 0 of set Y.

Composite Functions

Suppose we have two functions:

f\quad :\quad x\quad \rightarrow \quad { x }^{ 2 }   and
g\quad :\quad x\quad \rightarrow \quad { 2x\quad +\quad 3 }

We can combine these two functions to get a new function f\left[ g\left( x \right) \right] , and this new function is known as a composite function, i.e:

for the above two functions f\left[ g\left( x \right) \right] means:

=\quad f\left[ g\left( x \right) \right]

=\quad f\left( 2x\quad +\quad 3 \right)

Since:  f\left( x \right) \quad =\quad { x }^{ 2 }

f\left[ g\left( x \right) \right] ={ (2x\quad +\quad 3) }^{ 2 }

Inverse Function

Suppose we have a function f\left( x \right)\quad =\quad y, inverse of this function is the one that uses reverse relation i.e

f^{ -1 }\left( x \right) \quad =\quad y

However, remember that an inverse of a function will only exist if a function is one to one. Furthermore, if we draw the graphs of f\quad and\quad { f }^{ -1 } on the same axis, then the graph of { f }^{ -1 } will be the reflection of f in the line y = x as shown in Fig 1

Example #2

Q.    f\quad :\quad x\quad \rightarrow \quad \frac { 3x\quad +\quad 4 }{ 2 } ,\quad find\quad { f }^{ -1 }

Solution:

We know:

f\quad :\quad x\quad \rightarrow \quad \frac { 3x\quad +\quad 4 }{ 2 }

y\quad =\quad \frac { 3x\quad +\quad 4 }{ 2 }

To find inverse make x the subject of equation:

2y\quad =\quad 3x\quad +\quad 4

2y\quad -\quad 4\quad =\quad 3x

x\quad =\quad \frac { 2y\quad -\quad 4 }{ 3 }

Hence,

{ f }^{ -1 }\quad \rightarrow \quad x\quad =\quad \frac { 2y\quad -\quad 4 }{ 3 }       Ans

Modulus Function

A modulus is defined as an absolute value of x. It is basically a distance of x from a fixed point. It is written as \left| x \right| . A modulus function is defined as:

f\left( x \right)\quad =\quad \left| x \right| \left| x \right| \quad =\quad x\quad for\quad x\quad >\quad 0\quad and\quad \left| x \right| \quad =\quad -x\quad for\quad x\quad <\quad 0

Therefore the graph of y\quad =\quad \left| x \right|
is line y = x for x > 0 and y =  -x for x < 0 as shown below in Fig 2:

Graphical Transformations

Suppose we have a function y\quad =\quad f\left( x \right), some rules to remember when transforming graphs are:

  • The graph of y\quad =\quad f\left( x \right) \quad +\quad c where c is a constant, has the same shape of the function f\left( x \right) but is moved c units higher.
  • The graph of y\quad =\quad f\left( x \right) \quad -\quad c where c is a constant, has the same shape of the function f\left( x \right) but is moved c units lower.
  • The graph of y\quad =\quad f\left( x\quad +\quad k \right) , where k is a constant, has the same shape of the function but is moved k units to the left.
  • The graph of y\quad =\quad f\left( x\quad -\quad k \right) , where k is a constant, has the same shape of the function but is moved k units to the right.
  • The graph of y\quad =\quad f\left( -x \right) is simply a reflection of f\left( x \right)   in the y axis.
  • The graph of y\quad =\quad -f\left( x \right) is a reflection of f\left( x \right) in the x axis.
  • The graph of y\quad =\quad a\quad f\left( x \right) is stretched by a scale factor a in the y axis.
  • The graph of y\quad =\quad f\left( ax \right) is stretched by a scale factor \frac { 1 }{ a } in the x axis.
Reference
  1. https://www.mathedup.co.uk/key-stage-5/pure-maths/core-3/modulus-function/
  2. https://www.s-cool.co.uk/a-level/maths/functions/revise-it/transformations

 

 

Categories Algebra