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Definition of a function

A function is an expression of the form $$$ \Tred{f}(\Tblue{x}) = 2\Tblue{x}+5.$$$ It relates an input $$\Tblue{x}$$ to an ouput $$f(x)$$. The output is sometimes written as $$y$$. The function is $$\Tred{f}$$.

The value of the function $$\Tred{f}(\Tblue{x}) = 2\Tblue{x}+5$$ for the input $$\Tgreen{4}$$ is $$$ \Tred{f}(\Tgreen{4}) = 2\times \Tgreen{4} + 5 = 13.$$$

The input can be any number. It is represented by a letter, called the variable. The letter is generally $$\Tblue{x}$$, but it can be anything.

The functions $$\Tred{f}(\Tblue{t}) = \Tblue{t}^2$$ and $$\Tred{g}(\Tblue{x}) = \Tblue{x}^2$$ are the same, because the letters for the variable and for the function don't matter.

The function $$\Tred{f}(\Tblue{x}) = 2 \Tblue{x} -3$$ can be represented by a table for several inputs.

$$$x$$$ $$$-2$$$ $$$-1$$$ $$$0$$$ $$$1$$$ $$$2$$$ $$$3$$$ $$$4$$$
$$$f(x)$$$ $$$-7$$$ $$$-5$$$ $$$-3$$$ $$$-1$$$ $$$1$$$ $$$3$$$ $$$5$$$