Function (mathematics)

Mathematics is about the study of numbers and shapes, but it is also about the study of patterns and relationships. Function can define some of these relationships and so is an indispensable concept. Function is the core concept in calculus.

When learning about function students’ textbooks and teachers usually introduce function by way of a situation with the related quantities already identified. The students must learn how to set up and represent the relationships in tables, graphs, and equations. In the real world it is important to let students identify the quantities and let them determine which of these quantities may be related. The function is not the graph, not the table of values, and not the equation. The function is the relationship between the variables represented by these. The study of function is the study of these relationships and their properties. Identify the changing and unchanging quantities; determine the effect of the change of one quantity over the others; describe the properties of the relationship; and think of ways for describing and representing these relationships. These are the key concepts students should learn about function. Others concepts include looking at or dealing with functions as mathematical objects.

Constructing and Interpreting Linear Functions

Linear functions differ from exponential functions in their construction, and they serve different needs. For example, a team of scientists are studying the growth of two unknown pond weeds, called A and B. In the study, both pond weeds started at the height of 3 cm. Pond weed A seemed to triple in height every minute, whereas pond weed B grew at a constant rate of 3 cm every minute. So the series 3, 9, 27, 81, 243,... describes the growth of pond weed A: it is called exponential growth and can be represented by an exponential function. Whereas the series that describes the growth of pond weed B is 3, 6, 9, 12, 15,.... Growth is at a constant rate and can be modeled by a linear function.

Another story illustrates how to interpret linear equations. Mary and her father are going on a road trip. Mary’s father says that the journey is going to be a total of 300 miles long. On the first day they travel a distance of 50 miles; thereafter they drive a constant 50 miles every day thereafter. Mary wants to calculate how many days will pass before they reach their destination. The trip can be modeled by way of a linear function, of the form y = m.x + c. Specifically, 50 would be the y-intercept (c) and 50 would be the slope (m). So the linear function describing this trip is y = 50x + 50. Care is needed in the interpretation of this linear function. It might be thought that as 50 goes into 300 six times, the trip is going to take 300/50, that is, six days. This would, however, be ignoring the fact that there is an intercept of 50, so really the trip is going to take (300 − 50)/50 days, or 5 days.

Taylor Series of Basic Functions

Taylor series allow the mathematician to approximate functions that are otherwise difficult to calculate. The concept of a Taylor series was discovered by the Scot James Gregory but was formally introduced by the English mathematician Brook Taylor in 1715.

A Taylor series can be represented by f(x) = a₀ + a₁x + a₂x² + a₃x³ + .(extending to +ve ∞). An nth-degree Taylor polynomial for a function is the sum of the first n terms of a Taylor series. This is a finite series, and so can be computed exactly. Although it will not exactly match the infinite Taylor series or the original function, the approximation becomes progressively better as n increases. For example, the function y = sin(x) can be represented by the following Taylor polynomial:

As n becomes larger and there are more terms in the Taylor polynomial, the Taylor polynomial becomes a progressively better approximation of the function y = sin(x); it becomes more accurate. A Taylor polynomial is actually constructed according to the derivatives of the function under consideration at a certain point. Derivatives, roughly speaking, correspond to the shape of a curve, so the more derivatives that two functions have in common at one point, the more similar they will look at other nearby points.

Taylor series are important. They can be used to compute functions that cannot be computed directly. The above Taylor polynomial for the sine function looks complicated, but it is merely the sum of terms consisting of exponents and factorials, so the Taylor polynomial can be reduced to the basic operations of addition, subtraction, multiplication, and division. An approximation can be achieved by truncating the infinite Taylor series into a finite-degree Taylor polynomial, which can then be evaluated.

Inputs to a Function

When dealing with functions there are always three main parts: input, relationship, and output. A function relates an input to an output. The function f(2) = 8 simply means that somehow 2 is related to 8. A function takes elements of a set and gives back elements of another set. A set is simply a collection of things, for example, prime numbers or positive multiples of 3 that are less than 11. It is key to understand that a function must work for every possible input value, and that it has only one relationship for every input value.

Exponential Function

In exponential functions, the base is the fixed number, while the exponent is the variable (for example, y = 2^x ). Taking this function, as x increases by increments of 1from zero, y = 1, 2, 4, 8, and so on. As x decreases by increments of 1 from zero, y = 0.5, 0.25, 0.125, and so on, asymptotically approaching the x-axis. This is an example of exponential growth.

Exponential functions always have some positive number other than 1 as the base. Values 0 and 1 only give the same result: 0 and 1. Negative numbers oscillate; that is, negative even numbers give a positive result and odd numbers give a negative result. Powers that are not whole numbers then become an insurmountable problem.

One very important exponential function is the compound interest function.

Where A is the ending amount, P is the beginning amount (or principal), r is the interest rate (expressed as a decimal), n is the number of compoundings a year, and t is the total number of years. As n increases incrementally, A takes the following values: 2, 2.25, 2.441, 2.613, 2.693,...2.7182824. Here, the number e is the "natural" exponential, discovered by Leonard Euler. The function e^x is its own derivative. This function property leads to exponential growth and exponential decay, and e^x is a simple example of a Pfaffian function.

Domain and Range from Graph

Graphing any given function, the domain of the function is all x values for which the graph is valid. So for example any line, such as y = 3x + 9 can take any possible x value, positive or negative. The domain is then all x numbers, or you can say the domain is negative infinity to positive infinity.

But something an equation like y = sqrt(x) cannot take all numbers, because the square root of a negative number is not defined (as real). Therefore, the domain of this function is x >= 0, or you can say that the domain is 0 (inclusive) to infinity.

Sometimes certain numbers can be excluded. For example, y = 1/x is defined at all numbers except 0. The domain, then, is all numbers except 0, or negative infinity to 0 (not inclusive) and 0 (not inclusive) to infinity.

Evaluating Composite Functions

A composite function essentially has one function nested inside another. It can be written as f(g(x)), that is, f of g of x.

For example,

Always remember to put the inner function inside the outer function.

Bibliography

Garnett, John. Bounded Analytic Functions. New York: Springer, 2010.

Heins, Marice. Selected Topics in the Classical Theory of Functions of a Complex Variable. Dover, 2015.

Hurst, Edward, and Martin Gould. Bridging the Gap to University Mathematics. New York: Springer, 2009.

Newman, Mark E. J. "The Structure and Function of Complex Networks." Society for Industrial and Applied Mathematics Review 45.2 (2003).