What is Function?
Domain: The domain of the function is the set of all possible input values for which the function is defined. It's the "source" set. Every element in the domain must be used as an input to the function.
Range: The range of a function is the complete set of all possible output values, that the function can produce by applying its rule to the elements of its domain.
For a function f: A $\rightarrow $B, A is the domain, and the range is a subset of B.
Note: Domain restrict Division by zero, square root of negative numbers and similar things which does not exist.
Codomain: The codomain of a function f: A $\rightarrow $ B is the set B which contains all possible output values. It's the "target" set where the function's outputs are expected to land.
Range vs Codomain: The range is the actual set of output values produced by the function when applied to all elements of its domain (A). The range is always a subset of the codomain (Range $\subseteq $Codomain).
Think of it like this: if you're shooting arrows at a target (the codomain), the range is the set of holes you actually make on the target. You might aim for the entire target, but your arrows might only hit a specific section.
For example, if f(x) = $x^2 $with the domain being all real numbers ($\mathbb{R}$) and the codomain also being all real numbers ($\mathbb{R}$), the range is the set of all non-negative real numbers ($[0, \infty)$). Here, the range is a proper subset of the codomain.
However, if g(x) = $2x + 1$ with the domain and codomain both being all real numbers, then the range is also all real numbers because every real number can be obtained as an output. In this case, the range equals the codomain.
Therefore, the key difference is that the codomain is a declared set of potential outputs, while the range is the set of actual outputs achieved by the function. The codomain sets the stage, and the range is the reality of the function's output.
Operation of functions:
Given two functions, f(x) and g(x), we can define new functions based on the fundamental arithmetic operations:
Addition of Functions: The sum of two functions, denoted by (f + g)(x), is defined as the sum of their individual output values at a given input x:
(f + g)(x) = f(x) + g(x)
The domain of the resulting function (f + g)(x) is the intersection of the domains of f(x) and g(x). This is because for the sum to be defined at a particular x, both f(x) and g(x) must be defined at that x.
Example: If f(x) = $x^2 $(domain: all real numbers) and g(x) = $2x - 1 $ (domain: all real numbers), then
(f + g)(x) =$ x^2 + (2x - 1)$ =$ x^2 + 2x - 1$
The domain of (f + g)(x) is the intersection of ($-\infty, \infty$) and ($-\infty, \infty$), which is ($-\infty, \infty$).
Subtraction of Functions: The difference of two functions, denoted by (f - g)(x), is defined as the difference of their individual output values at a given input x:
(f - g)(x) = f(x) - g(x)
Similar to addition, the domain of (f - g)(x) is the intersection of the domains of f(x) and g(x).
Example: Using the same f(x) and g(x) as above,
(f - g)(x) = $x^2 - (2x - 1)$ =$ x^2 - 2x + 1$
The domain of (f - g)(x) is also ($-\infty, \infty$).
Multiplication of Functions: The product of two functions, denoted by ($f \cdot g$)(x) or (fg)(x), is defined as the product of their individual output values at a given input x:
($f \cdot g$)(x) = $f(x) \cdot g(x)$
Again, the domain of ($f \cdot g$)(x) is the intersection of the domains of f(x) and g(x).
Example:
($f \cdot g$)(x) = $x^2 \cdot (2x - 1)$ =$ 2x^3 - x^2$
The domain remains ($-\infty, \infty$).
Division of Functions: The quotient of two functions, denoted by ($f / g$)(x), is defined as the ratio of their individual output values at a given input x:
(f / g)(x) = $\frac{f(x)}{g(x)}$
The domain of (f / g)(x) is the intersection of the domains of f(x) and g(x), with the crucial additional restriction that $g(x) \neq 0$ for any x in the domain. We must exclude any x values where the denominator becomes zero to avoid undefined results.
Example:
(f / g)(x) = $\frac{x^2}{2x - 1}$
The domain of f(x) is ($-\infty, \infty$), and the domain of g(x) is ($-\infty, \infty$). However, g(x) = 0 when 2x - 1 = 0, which means x = 1/2. Therefore, the domain of (f / g)(x) is all real numbers except 1/2, which can be written as $(-\infty, 1/2) \cup (1/2, \infty)$.
Composition of Functions:
Another fundamental operation on functions is composition. This involves applying one function to the result of another function.
The composition of a function f with a function g, denoted by ($f \circ g$)(x) or f(g(x)), is defined as:
($f \circ g$)(x) = f(g(x))
This means we first evaluate g(x), and then we use the result as the input for the function f.
Similarly, the composition of g with f, denoted by ($g \circ f$)(x) or $ g(f(x))$, is defined as:
($g \circ f$)(x) = g(f(x))
In this case, we first evaluate f(x), and then use the result as the input for the function g.
Important Note: In general, $f(g(x)) $is not the same as $ g(f(x))$. The order of composition matters!
The domain of the composite function $(f \circ g)(x) $ consists of all x in the domain of g such that g(x) is in the domain of f. In other words, the inner function g(x) must produce an output that the outer function f can accept as an input.
Example 1: Let f(x) =$ \sqrt{x}$ (domain: $[0, \infty)$) and g(x) = x - 2 (domain: $(-\infty, \infty)$).
$(f \circ g)(x) $= $f(g(x)) $= f(x - 2) = $\sqrt{x - 2}$. The domain of ($f \circ g$)(x) requires $x - 2 \ge 0$, so $ x \ge 2$. The domain is [$2, \infty$).
($g \circ f$)(x) = $g(f(x)) $=$ g(\sqrt{x}) $= $\sqrt{x} - 2$. The domain of ($g \circ f$)(x) is the domain of f(x), which is [$0, \infty$).
Injection function(One-to-one): A function f:$ A \rightarrow $B is injective if every distinct element in the domain A maps to a distinct element in the codomain B. In other words, if $f(x_1)$ =$ f(x_2)$, then $x_1$ =$ x_2$. Each output has at most one input.
Surjection function (Onto): A function f: $A \rightarrow $B is surjective if every element in the codomain B is the image of at least one element in the domain A. That is, for every y $\in B$, there exists at least one x $\in A $ such that f(x) = y. The range of the function equals the codomain.
Bijection function(One-to-one and Onto): A function f: A $\rightarrow$ B is bijective if it is both injective and surjective. This means there is a perfect pairing between the elements of the domain and the codomain; each element in A maps to a unique element in B, and every element in B has exactly one element in A that maps to it. Bijective functions have inverse functions.
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