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1.1: Real Functions

Last updated

Apr 30, 2021
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- 1: Mathematical Functions
- 1.2: 1.2 The Exponential Function

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A mathematical function, denoted $f$ , takes an input $x$ (which is also called an argument), and returns an output $f(x)$ . For now, we consider the case where both $x$ and $f(x)$ are real numbers. The set of possible inputs is the function’s domain, and the set of possible outputs is the range.

Every function must have a well-defined output: for any $x$ in the domain, $f(x)$ must be a specific, unambiguous number. In other words, a function must be either a one-to-one (injective) mapping or a many-to-one mapping; the mapping cannot be one-to-many or many-to-many:

Simple examples of functions are those based on elementary algebra operations:

$\begin{align} f(x) &= x + 2 \,\;\;\qquad\qquad \text{(a one-to-one function)} \\ f(x) &= x^2 + 2x + 4 \qquad \text{(a many-to-one function)}\end{align}$

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