07. Operations on Functions
Dated: 30-10-2024
Operations
Just like numbers, we can do operations on functions1 as well.
Assume we have 2 functions1 such as:
\[f(x) = x^2\]
\[g(x) = x + 1\]
Addition
\[f(x) + g(x) = (f + g)(x) = x^2 + x + 1\]
Subtraction
\[f(x) - g(x) = (f - g)(x) = x^2 - (x + 1)\]
Multiplication
\[f(x) \cdot g(x) = (f \cdot g)(x) = x^2 \cdot (x + 1)\]
Notation
\[f^n(x) = \prod^n f(x) = f(x) \times f(x) \times \ldots \times f(x)\]
Division
\[\frac{f(x)}{g(x)} = \left(\frac{f}{g}\right)(x) = \frac{x^2}{x + 1}\]
The domain2 of these functions1 which result after operations, is the intersection3 of the domains2 of \(f(x)\) and \(g(x)\).
Basically saying that we need certain set4 of values which work for both functions.1
Composition
\[f(g(x)) = (fog)(x) = (x+1)^2\]
\[g(f(x)) = (gof)(x) = x^2 + 1\]
Therefore,
\[(fog)(x) \ne (gof)(x)\]
Decomposition
Similarly, functions1 can be decomposed as well.
\[h(x) = (x+1)^2\]
\[h(x) = (fog)(x) = f(g(x))\]
Classification of Functions
Constant
The functions1 whose range5 is equal to a constant number is called a constant function.
Monomial in \(x\)
Anything of the form
\[f(x) = cx^n ; x \in \mathbb{Z}^+ + \{0\}\]
Polynomial in \(x\)
Anything of the form
\[f(x) = \sum_{i = 0}^n a_ix^i = a_0 + a_1x^1 + \ldots + a_nx^n\]
References
Read more about notations and symbols.