A function of numbers has a name, a rule and a domain. The name can be any string of characters, but it is best if it is short and somehow memorable. If all you know about the function is its rule in arithmetic form, then it is customary to use f, g, h etc. for the name. Similarly, an element of the domain might have a name related to its units , if applicable. For example, the function whose value is the distance associated with a time in hours traveled by a vehicle moving at 45 miles per hour: Dist(hr) or d(t). As to our study of algebra, the names are insignificant, but sometimes convenient mnemonic devices.
The domain is sometimes specified, like “all the positive numbers” but more often it defaults to all real numbers for which the function is well-defined, i.e., valid.
The rule that associates a “value” in the domain with a real number is typically a numeric expression with a place-holder name for the argument (element of the domain). The function of a specific argument is evaluated by substituting that argument for the place-holder.
F(x) = 2x² – 3x + 5
F(-2) = 2((-2)² – 3(-2) + 5 = 8 + 6 + 5 = 19