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*) = 3*x* + 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*) = *ax*^{2} + *bx* + *c*, for constants *a*, *b* and *c*. If *a* = 2, *b* = 3 and *c* = 1, the equation becomes *f*(*x*) = 2*x*^{2} + 3*x* + 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.