Functions: Basic Concepts

A function in mathematics can be defined as a relation between a set of inputs & a set of outputs with a certain property that each input is related to one output. Example: A function that relates each real number x to its cube x4. The output of a function f relating to an input x is presented as f(x). Let us say, if the input in this case is -2, then the corresponding output will be f(-2) = 16. Another example, if the input is 2, the output will be f(2) = 16. (In this case, we can see that the same output can be produced by more than one input, but each input gives a unique output.)
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Types of functions:
  • Odd Functions: A function is an odd function if & only if f(x) = -f(-x). The graph of an odd function is said to be symmetrical about the alternate quadrants of the graph. E.g. y = x3
  • Even Functions: A function is an even function if f(x) = f(-x) that is the value of that particular function remains same if if we replace x by –x. The graph of an even function is symmetrical about y axis example: y = x2.
Note: A function is said to be neither even nor odd, if it does not satisfy any of the two given conditions above.
Examples of functions:
Function example 1:
If f(x) = 3x + 1 & g (x) = x2 + 4
Find f(2) –g(2)
Now, f(2) = 3(2) + 1 = 6 + 1 = 7
g(2) = 22 + 4 = 4 + 4 = 8
Therefore, f(2) – g(2) = 7 – 8 = -1
Function example 2:
f(x) = 4x ; if x is odd
f(x) = x2 + 1 ;if x is even
Find f(f(3))
Now, in first case f = 3 which is odd. Therefore, f (4(3)) = f(12)
Now x = 12, which is even, therefore, f(12) = (12)2 + 1 = 144 + 1= 145
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