What is Function Notation?

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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​) = ​ax2 + ​bx​ + ​c​, for constants ​a​, ​b​ and ​c​. If ​a​ = 2, ​b​ = 3 and ​c​ = 1, the equation becomes ​f​(​x​) = 2​x2 + 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

y = ax^n +bx^{(n − 1)} +cx^{(n − 2)} + ...

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

y^2 = x

is not a function for dependent variable ​y​. Re-writing the equation it becomes

y = \sqrt{x}

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

y = ax^2 + bx + c

for constants ​a​, ​b​ and ​c​ is a function and can be written as

f(x) = ax^2 + bx + c

If ​a​ = 2, ​b​ = 3 and ​c​ = 1, this becomes:

f(x) = 2x^2 + 3x + 1

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.

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