Compositioninverse
Afunction is a rule that assigns to each input number exactly oneinput number. The set of all input numbers to which the rule appliesis called the domain of the function. The set of all output numbersis called the range [ CITATION Anv80 l 1033 ].
Twofunctions can be combined to form a new function resulting incomposition or a composite function. Several operations can be doneon the function, which include drawing a graph for the function,translations, finding the inverse of the function and many otheroperations. A few cases of such manipulations with functions aredemonstrated in this paper.
Method

Two functions are given:
f(x)= 2x + 5 and g(x) = x^{2}– 3. A new function can be formed from these two functions asfollows:
h(x)= g(f(x)) = g(2x + 5) :function f substituted in g
h(x)= (2x + 5)^{2}– 3 :h(x) evaluated as a function of another function
=4x^{2}+ 20x + 25 – 3
Therefore,h(x) = 4x^{2}+ 20x + 22
Tocompute (f – h) (4), the expression can be rewritten as f(4) –h(4)
=[(2×4) + 5] – [4×16 + 20×4 + 22] :4 is substituted for x in bothfunctions
=13– 166
=153
Functionscan be combined to give compositions [ CITATION SOS14 l 1033 ]. Evaluatingthe compositions given can be done as shown below.

(f o g)(x) = f(g(x)) = f(x^{2} – 3)
=2(x^{2}– 3) + 5
=2x^{2}– 6 + 5
Therefore(f o g)(x) = 2x^{2}– 1

(h o g)(x) = h(g(x)) = h(x^{2} – 3)
=4(x^{2}– 3)^{2}+ 20(x^{2}– 3) + 22
=4(x^{4}– 6x^{2}+ 9) + 20x^{2}– 60 + 22
=4x^{4}– 24x^{2}+ 36 +20x^{2}– 60 + 22
=4x^{4}– 4x^{2}– 2
Therefore,(h o g) (x) = 4x^{4}– 4x^{2}– 2.

The function f(x) = 2x + 5 implies that y = f(x) = 2x + 5. The working below shows how we can get an inverse for this function [ CITATION Pur14 l 1033 ]
y– 5 = 2x
x= (y – 5)/2
f^{1}(x)= (x – 5)/2 :this is the inverse function after substituting x fory.

For h(x) = 4x^{2} + 20x + 22 = y, the expression can be written as
4x^{2}+ 20x + (22 – y) = 0
Usingthe quadratic formula, values for a = 4, b = 20 and c = 22 – y.
Hence x = [20 ± √ (400 – 16(22 – y)]/ 8
=[20 ± 4√ (25 – (22 – y)]/ 8
=[20 ± 4√ (3 + y)]/ 8
=[5 ± √ (3 + y)]/ 2
Fromthe working above, the inverse of the function h(x) is
h^{1}(x)= [5 ± √ ( 3 + x)]/ 2

Given the function g(x) = x^{2} – 3, the table for the domain and range of the function including a translation of 6 units to the right and 7 units down is,

X
6
5
4
3
2
1
0
1
2
3
4
5
6
g(x)
33
22
13
6
1
2
3
2
1
6
13
22
33
X+6
0
1
2
3
4
5
6
7
8
9
10
11
12
g(x)7
26
15
6
1
6
9
10
9
6
1
6
15
26
Thegraph of the g(x) function is given below:
Thenext graph gives g(x) when translated or transformed by 6 units tothe right and 7 units down.
Conclusion
Functionsare versatile in terms of the operations that can be performed onthem. It is possible also to combine functions and use one unifiedequation to carry out the operations which would yield the sameresults as the one given by separate equations of the functions.
References
Anvil, E.a. (1980). Analytical mathematics. London: OUP.
Purple Math. (n.d.). Retrieved May 22, 2014, from purple math web site: http://www.purplemath.com/modules/fcncomp6.htm
SOS math. (n.d.). Retrieved May 22, 2014, from SOS math web site: http:www.sosmath.com/algebra/invfunc/fnc3.html