Function notation is a compact form used to express the dependent variable of a function in terms of the independent variable. Using function notation, y is the dependent variable and x is the independent variable. The equation of a function is y = f(x), which means y is a function of x. All the independent variable x terms of an equation are placed on the right side of the equation while the f(x), representing the dependent variable, goes on the left side.
If x is a linear function for example, the equation is y = ax + b where a and b are constants. The function notation is f(x) = ax + b. If a = 3 and b = 5, the formula becomes f(x) = 3x + 5. Function notation permits the evaluation of f(x) for all values of x. For example, if x = 2, f(2) is 11. Function notation makes it easier to see how a function behaves as x changes.
TL;DR (Too Long; Didn't Read)
Function notation makes it easy to calculate the value of a function in terms of the independent variable. The independent variable terms with x go on the right side of the equation while f(x) goes on the left side.
For example, function notation for a quadratic equation is f(x) = ax2 + bx + c, for constants a, b and c. If a = 2, b = 3 and c = 1, the equation becomes f(x) = 2x2 + 3x + 1. This function can be evaluated for all values of x. If x = 1, f(1) = 6. Similarly, f(4) = 45. Function notation can be used to generate points on a graph or find the value of the function for a specific value of x. It is a convenient, shorthand way to study what a function's values are for different values of the independent variable x.
How Functions Behave
In algebra, equations are generally of the form
where a, b, c ... and n are constants. Functions may also be predefined relations such as the trigonometric functions sine, cosine and tangent with equations such as y = sin(x). In each case, functions are uniquely useful because, for every x, there is only one y. This means that when the equation of a function is solved for a particular real life situation, there is only one solution. Having a single solution is often important when decisions have to be made.
Not all equations or relations are functions. For example, the equation
is not a function for dependent variable y. Re-writing the equation it becomes
or, in function notation, y = f(x) and f(x) = √x. For x = 4, f(4) can be +2 or −2. In fact, for any positive number, there are two values for f(x). The equation y = √x is therefore not a function.
Example of a Quadratic Equation
The quadratic equation
for constants a, b and c is a function and can be written as
If a = 2, b = 3 and c = 1, this becomes:
No matter what value x takes, there is only one resulting f(x). For example, for x = 1, f(1) = 6 and for x = 4, f(4) = 45.
Function notation makes it easy to graph a function because y, the dependent variable of the y-axis is given by f(x). As a result, for different values of x, the calculated f(x) value is the y-coordinate on the graph. Evaluating f(x) for x = 2, 1, 0, −1 and −2, f(x) = 15, 6, 1, 0, and 3. When the corresponding (x, y) points, (2, 15), (1, 6), (0, 1), ( −1, 0) and ( −2, 3) are plotted on a graph, the result is a parabola shifted slightly to the left of the y-axis, passing through the y-axis when y is 1 and passing through the x-axis when x = −1.
By placing all the independent variable terms containing x on the right side of the equation and leaving f(x), which is equal to y, on the left side, function notation facilitates a clear analysis of the function and the plotting of its graph.