Overview: What Are Functions?
Functions in mathematics describe two different types of relationships between numbers. The simplest definitions are that a function can be either the set of ordered pairs in which the first element is paired with a second element or the relationship between two sets of elements in which every element in Set A can be paired with an unique element in Set B. The definitions are different from each other to reflect the many ways that functions can be represented — as pairs in tables or lists, in rules made up of words, or in graphs. The first elements in the ordered pair or in Set A are the domain or independent variable, represented on the x-axis; and the second elements in the ordered pair or in Set B are the range or dependent variable, represented on the y-axis.
Linear Functions
Linear functions follow the general form y= mx +b, where m is a constant representing the slope and b is a constant representing the y-intercept (the point at which the line crosses the y-axis). In the real world, most data sets do not follow an exact line, but a line of best fit can be approximated using regression techniques. One common type of linear function used frequently in the social sciences, the correlation coefficient, measures how strongly one variable is related to another.
Quadratic Functions
Quadratic functions are one type of nonlinear model in which the general form of the equation of best fit follows the quadratic formula ax2 + bx +c = 0, where a is not equal to 0, and a, b, and c are all constants. The resulting graph is a parabola. There are many real-world functions that are quadratic, including the Newtonian measurement of the relationship between gravity, velocity, and time. In the equation the variable h represents height the object is initially thrown from, g is the acceleration constant due to gravity, v is velocity, and t is time, so that h=-1/2gt2-v0t +h0.
Exponential and Logarithmic Functions
An exponential function occurs when the form of the function f(x) = abx, when a is not equal to 0, b is greater than 0 and not equal to 1. (If b is equal to 1, then the function is linear.) Many different types of exponential functions, such as population growth, occur in the natural world. Logarithmic functions are solved for y rather than x, so they are the inverse. They also form a family of curves.
Trigonometric Functions
Trigonometric functions include circular functions such as sines, cosines, and tangents, which are derived from the relationships between angles and sides within triangles. As with the graphs of quadratic, exponential, and logarithmic functions, the graphs of trigonometric functions show consistent patterns. The wave patterns illustrate periodic events such as sound and light waves.
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