Real Function/Examples

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Examples of Real Function

Square Function

The (real) square function is the real function $f: \R \to \R$ defined as:

$\forall x \in \R: \map f x = x^2$

Square Root Function

The (real) square root function is the real function $f: \R \to \R$ defined on the positive real numbers as:

$\forall x \in \R_{\ge 0}: \map f x = \sqrt x$

Arbitrary Real Function $1$

Let $f$ denote the real function defined as follows:

$\map f x = \begin {cases} 2 & : 0 \le x \le 1 \\ \sqrt x & : 2 \le x \le 3 \end {cases}$

Thus the domain of $f$ is seen to be:

$\Dom f = \closedint 0 1 \cup \closedint 2 3$


Arbitrary Real Function $2$

The following defines a real function:

$\forall x \in \R: \map f x = 6 + x^3$

Arbitrary Temperature

The following is an invented graph of a hypothetical output from an automatic temperature sensing device over the period of $12$ hours.


Note that while there is no way to express the temperature as a formula, the relation between the time and the temperature is still a function.