With so many formulas and equations to know and use, computations can become confusing when you must use several formulas and/or equations for the same problem. Function notation was developed to help differentiate between distinct equations and formulas, which are all considered functions.
A function is a correspondence, or relation, between two sets of data where each input (one set of data) is associated with one output (second set of data). In other words, inputting a specific value into an equation or formula will result in one specific value. If an input results in multiple outputs, the equation or formula cannot be considered a function. This occurs when there are 2 y-values for a single value of x on the graph of the function.
Function notation refers to replacing the output variable (usually y) with a different notation: f(x). This notation is not f multiplied by x, but is simply read as “f of x.” The f is a label or a way to distinguish which equation is which. You can use whatever letter you want for function notation, such as g, q, or z, but most functions will be written with the f. The variable inside of the parentheses is the input.
A composite function is a function of another function, and it is where a function is plugged in for x. This creates a function such as f(g(x)), or “f of g of x.” We can solve these by substituting the function inside the parentheses for the x in the function outside the parentheses.
You will usually not get the same answers for f(g(x)) and g(f(x)), so make sure you are composing functions in the correct order!
Answers to Practice Problems