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# Function

Let A and B be the two non-empty sets. A relation from set A to set B is said to be a function if every element of set A has unique element in set B (only one element in set B).
Symbolically: for every x $\in$ A, there exist an element y $\in$ B, such that
y = f(x) $\in$ B.

Algebraic Function
An algebraic functions are those functions which involve sum, difference of algebraic terms. The types of algebraic functions are:

1. Constant Function
2. Linear Function
3. Identity Function
5. Polynomial Function
6. Cubic Function

Trigonometric Function
A function f:A$\rightarrow$ B is said to be a trigonometric function if the function f involve trigonometric ratios: sine, cosine, tangent, etc.
y = sin x, y = cos x, y = sin x + cos x etc.

Onto Function
A function f: A $\rightarrow$ B is said to be an onto function if the range set is equal to the co-domain i.e. every image has pre image.

Into Function
A function f: A $\rightarrow$ B is said to be an into function if the range is subset of co-domain i.e. some of the elements on the set B are present without any pre image.

Inverse Function
Let f:A $\rightarrow$ B be a one to one function, then function g: B $\rightarrow$ A such that for each x $\in$ B there exist g(x) $\in$ A is called inverse function of f. It is denoted by g = f-1.
Note: If f: A $\rightarrow$ B be one to one onto function then interchanging the domain and range of function is the inverse of function f. While finding the inverse function, the role of domain and range will be interchanged.
E.g. f = {(2, 4), (3 , 6), (5, 10)}
f-1 = {(4, 2), (6, 3), (10, 5)}

Example:
If f: R $\rightarrow$ R defined by f(x)= 5x – 3, find f-1.
Solution:
Here, x is pre-image and y is image of f. y is pre-image and x is image of f-1.
f(x)= 5x – 3
Let y = 5x – 3
Interchanging x and y
x = 5y – 3
or, x + 3 = 5y
or, y = $\frac{x + 3}{5}$
$\therefore$ f-1(x) = $\frac{x + 3}{5}$
Alternatively,
f (x) = 5x – 3
or, y = 5x – 3
or, y + 3 = 5x
or, $\frac{y + 3}{5}$ = x
$\therefore$ f-1(x) = $\frac{y + 3}{5}$
[Note: Inverse function is possible only on one to one onto function.]

Composite function
Let f be a function from A to B and g be a function from B to C. Then the function defined from A to C is the composite function. Composite function is also known as the function of function.
E.g. Paddy $\rightarrow$ Rice mill $\rightarrow$ Rice $\rightarrow$ Flour Mill $\rightarrow$ Flour $\rightarrow$ Paddy
Let f: A $\rightarrow$ B and g: B $\rightarrow$ C be two functions. Then the new function defined from A to C such that every element of A corresponds with a unique element of C is known as composite function of f and g. it is denoted by gof(x) or gf(x).

[Note: Inverse function and composite function are very much important for the exam point of view.]

Examples:

1. If f = {(1, 4) (2, 0) (-1, -4)} and g = {(0, 3) (4, 6) (-4, -3)}, form the ordered pairs for gof and represent them in mapping diagram.
Solution:
Here,
f(x) = {(1, 4) (2, 0) (-1, -4)}
g(x) = {(0, 3) (4, 6) (-4, -3)}
$\therefore$ gof(x) = {(1, 6) (2, 3) (-1, -3)}
2. If f = {(2, 4) (3, 6) (4, 8) (2, 12)} and g = {(6, 2) (4, 12) (8, 16) (12, 6)} form ordered pairs for gof and represent in mapping diagram.
Solution:
Here,
f = {(2, 4) (3, 6) (4, 8) (2, 12)}
g = {(6, 2) (4, 12) (8, 16) (12, 6)}
gof = {(2, 12) (3, 2) (4, 16) (6, 6)}

Exercise

1. If f(x) = 5x + 2, g(x) = 3x – 3, find
(i) fog(x)
(ii) gof(x)
2. If g(x) = 7 – 3x, h(x) = 3x – 5, find
(i) g-1h
(ii) (hog)-1(3)
3. If f(3x + 2) = 9x + 10, find f-1(3).
4. If f-1(x) = x – 2, find f(x) and f(-5).
5. If f(x) = 3x + a and ff(6) = 10, find the value of a.
6. If f= (2, 4) (1, 2) (3, 5), g(x) = (2, 7) (4, 9) (5, 10), then find gof in an arrow diagram.
7. If function f(x) = 2x – 4, then prove that fof-1(x) is an identity function.
8. If f(x) = 3x + 4 and g(x) = $\frac{4x -13}{2}$. If ff(x) = g-1(x), find the value of x.
9. If function f(x) = x + a, g(x) = 2x – a and gf(4) = 11, find the value of a.
10. If function f(x) = $\frac{2x + 3}{2}$ and g(x) = $\frac{3x – 2}{4}$, find the value of g-1(x) and fg-1(3).