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Recall that a function is any rule which assigns a unique value in the range of the function to any value in the domain of the function. It is common to encounter functions which can be expressed through simple algebraic equations, but this is not the only way to define functions.
To get you started you might think about these sorts of functions: polynomials, trig functions, exponentials, integer powers of the variable and decide in which function categories these belong.
You might then consider some more complicated functions. For example, some might be defined in two parts such as $f(x) = A(x), x\geq 0$ and $f(x) = B(x), x<0$.
Don't be afraid to be inventive!