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We often need the concept of functions in probability. A function $f$ is a rule that takes an input from a specific set, called the domain, and produces an output from another set, called co-domain. Thus, a function maps elements from the domain set to elements in the co-domain with the property that each input is mapped to exactly one output. For a function $f$, if $x$ is an element in the domain, then the function value (the output of the function) is shown by $f(x)$. If $A$ is the domain and $B$ is the co-domain for the function $f$, we use the following notation: $$f:A \rightarrow B.$$

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Example

• Consider the function $f:\mathbb{R} \rightarrow \mathbb{R}$, defined as $f(x)=x^2$. This function takes any real number $x$ and outputs $x^2$. For example, $f(2)=4$.
• Consider the function $g:\{H,T\} \rightarrow \{0,1\}$, defined as $g(H)=0$ and $g(T)=1$. This function can only take two possible inputs $H$ or $T$, where $H$ is mapped to $0$ and $T$ is mapped to $1$.

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The output of a function $f: A \rightarrow B$ always belongs to the co-domain $B$. However, not all values in the co-domain are always covered by the function. In the above example, $f:\mathbb{R} \rightarrow \mathbb{R}$, the function value is always a positive number $f(x)=x^2 \geq 0$. We define the range of a function as the set containing all the possible values of $f(x)$. Thus, the range of a function is always a subset of its co-domain. For the above function $f(x)=x^2$, the range of $f$ is given by $$\textrm{Range}(f)=\mathbb{R}^{+}=\{x \in \mathbb{R} | x \geq 0 \}.$$

Figure 1.14 pictorially shows a function, its domain, co-domain, and range. The figure shows that an element $x$ in the domain is mapped to $f(x)$ in the range.

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