Graphing of Functions
Graphing of functions is a fundamental method in mathematics that allows for the visualization of relationships between sets of inputs and outputs. A function is defined as a rule that assigns each input from a specific domain to a single output within a corresponding range. This relationship can be represented graphically in various coordinate systems, with the Cartesian coordinate system being the most common for two-dimensional graphs. Here, an input value \( x \) leads to an output value \( f(x) \), which can be plotted as an ordered pair on the graph.
In addition to two-dimensional graphs, functions can also be extended to three dimensions, where they take two inputs, \( x \) and \( y \), and produce a single output \( f(x, y) \), representing a surface. Notably, not every relationship qualifies as a function; a relationship must adhere to the condition that each input corresponds to only one output. The vertical line test is a useful tool for determining if a relationship is a function: if any vertical line intersects the graph at more than one point, it fails this test. Understanding how to graph functions is essential for analyzing mathematical patterns and real-world phenomena, from physics to economics.
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Graphing of Functions
Developing the proper methods to visualize concepts in mathematics is critical, as it provides simplistic approaches in discovering and analyzing patterns. Functions are an elementary example of a way to think about concepts that may be represented visually. A function takes a set of inputs and outputs a single value. The domain of a function is defined as the set of values that can be inputted into that function, whereas the range is the set of values that can be outputted. For function notation in two dimensions, f(x) describes the function accepting a single input x, and outputting a value f(x). Thus the values (x, f(x)) may be represented as an ordered pair as such, making them possible to graph on a particular coordinate system.
Functions can describe phenomena in the real world, from the relationship between the electrical current traveling through a DC circuit over time, to the gravitational force between two bodies in relation to their distance from one another. The role functions play are, therefore, essential building blocks in that they provide tools to understand the language of mathematics.
Overview
When graphing functions, the values must lie in a particular coordinate system. The most familiar of such is called a Cartesian coordinate system, in which each of the respective axes are perpendicular to one another. In two-dimensional functions, the function f(x) will mark the input x value, and plot a corresponding f(x) value (y value) for that particular point. In three dimensions, the function will take two inputs, x and y, and output a single value f(x, y), often to represent a three-dimensional surface. The same logic applies to an n dimensional surface, although these are impossible to visualize because the world is defined in three dimensions.
The function f(x) = x visually depicts that of a straight line rotated 45 degrees and passing through the origin of the Cartesian plane. The domain of this function is the set of all real numbers, since there is no real number that makes the function undefined. The range of the function is also in the same set, since the set of all real numbers may be outputted. The function 1/x, on the other hand, is undefined when x = 0 (division by zero is undefined); therefore, the domain is restricted only at x = 0.
Not all mathematical relationships that can be graphed define a function, however. A relationship is only defined a function if on the region in which the function is defined, there exists only one function value that maps to a particular x value. Visually, if one was to construct a relationship in which two f(x) or y values were on the same x position, one could draw a vertical line between these two points. If at any point along the graph one of these vertical lines exists, the relationship is not a function. This specific test is called the vertical line test. When a relation fails to classify as a function, it is said to be in implicit form, whereas, when one represents a function, that function is in explicit form.
Bibliography
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Kaufmann, Jerome E., and Karen L. Schwitters. Algebra for College Students. Belmont, CA: Cengage, 2011.
Larson, Ron, and David C. Falvo. Precalculus. Boston: Cengage, 2014.
Ostermann, Alexander, and Gerhard Wanner. Geometry by Its History. New York: Springer, 2012.