Question

What are the differences among expressions, equations, and functions? Provide examples of each

Answer 1

Expressions are just a group of term(s). eg. 3xy - 16x^2
Equations are expressions connected by an 'equals' sign eg. 45t - 16 = 29, xy = 16
Functions are machines that convert inputs from a set of elements (called the domain) into outputs belonging to a set called the codomain (or range). You can think of them as operators. Be aware that f(x) = x^2 is NOT a function. f is the function, and f(x) is the function evaluated at the input x, and f(x) = x^2 is an equation.
The function does not have a value on its own. 'To square', 'Add 5', 'Take the squareroot of', 'differentiate' are all functions. They tell what to do with the input. Sometimes, this 'operator' is not mentioned explicitly. That being said, any set of ordered pairs such that each input has exactly one unique output is considered a function.
eg. Let f be a function, given by the equation f(x) = x^2, for all real x.