From the course: Complete Guide to Calculus Foundations for Data Science

Polynomial and rational functions

From the course: Complete Guide to Calculus Foundations for Data Science

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Polynomial and rational functions

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- [Instructor] First up are polynomial and rational functions. Before you get into those though, let's begin by understanding what algebraic functions are. Algebraic functions are defined by the root of a polynomial equation with one or more variables. These functions are made from algebraic operations such as addition, subtraction, multiplication, division, and exponentiation. Let's look at a few examples of functions that are algebraic and ones that are not algebraic. To the left, you have some algebraic functions of f of x equals the square root of x squared plus four, and then you have f of x equals x to the fifth minus three multiplied by x squared plus seven, and finally, you have f of x equals the cube root of x to the third plus two multiplied by x plus one. All three of these functions on the left are algebraic because they are found using those algebraic operations. To the right, you have some non-algebraic functions where you have f of x equals x to the x. This is not algebraic because you are taking the variable in the exponent. Next, you have f of x equals sine of x and f of x equals ln f x, which is the natural log of x. These two are imposing some sort of other function operation on them, such as a trigonometric or a logarithmic function, hence, they are not algebraic. There are various types of functions that are derived from algebraic functions, such as polynomial functions and rational functions. Let's begin exploring polynomial functions. These functions are in the form of an multiplied by x to the n plus an minus one multiplied by x to the n minus one plus all the way to plus a1 multiplied by x plus a0. Note that in this case, n must be greater than zero, and a of n must not equal zero. Let's take a look at the degrees of polynomials. So in this case, the degree of the polynomial is represented by the variable n. So if you have the degree of n equal to zero, then it is a constant function where it equals constant values such as seven or 29. If the degree n equals one, then it is a linear function. If the degree is n equals two, then it is a quadratic function. And finally, if the degree n equals three, it is a cubic function. Let's look at a few examples of polynomial functions. For f of x, you have x squared plus one, which this would be a quadratic function since it has a squared term. Next, you have f of x equals x to the fourth minus five multiplied by x to the third plus eight multiplied by x squared minus x. Finally, you have f of x equals seven, which is a constant function because the degree of n equals zero, and hence, it's like multiplying x to the zero, which just equals one. To the right, you will see what a graph may look like for a polynomial function, where oftentimes they are represented by various curves such as different U's, or sometimes they kind of come up from the left side and then up to the right or vice versa. Let's take a further look at the degrees of polynomials and what they can tell you. Something interesting is how the sine and degree of the leading coefficient of a polynomial function can determine the behavior of its graph. So in the first case, if you have an even degree and a positive leading coefficient, then both ends of the graph will rise toward infinity. Number two, if you have an even degree and negative leading coefficient, then both ends of the graph will fall towards negative infinity. Number three, if you have an odd degree and a positive leading coefficient, then the left end of the graph falls, which will be towards negative infinity and the right end of the graph rises, which will be towards positive infinity. And finally, number four, if you have an odd degree and negative leading coefficient, the left end of the graph will rise towards positive infinity, and the right end of the graph will fall towards negative infinity. There are many properties for polynomial functions, but let's review some of the key ones. First, you have the roots are values of x, where f of x equals zero. You'll hear roots mentioned many times throughout this course, so it's very important to know that you can find them using this method. Number two, a polynomial function with a degree of n can have up to n real roots. Number three, polynomial functions are continuous and differentiable for all real values of x. Number four, a polynomial function with a degree of n can have up to n minus one turning points. This means where the graph will change direction. Number five, polynomial functions do not have vertical, horizontal, or oblique asymptotes. And finally, number six, polynomials can sometimes exhibit symmetry if they contain all odd or all even powers of x. Let's pivot and take a look at rational functions. Rational functions are where you have a function f of x in the form of f of x equals g of x divided by h of x. In this form, g of x and h of x are both polynomial functions. And note that h of x must not equal zero because you cannot divide a fraction by zero and have it be a valid result. Hence, rational functions are basically a quotient of polynomial functions, making them extremely related to them. Let's take a look at what some rational functions may look like. First you have f of x equals three multiplied by x to the fifth minus two multiplied by x squared plus three, and then you'll divide all of that by x squared minus one. So this is a rational function because you have your two polynomial functions on the top and bottom portion of your fraction. Next, you have f of x equals x squared divided by x cubed minus two multiplied by x. Again, you have your two polynomial functions, even if you just have that single term of x squared on the top. And finally, you have f of x equals three divided by x minus four. Remember that even though this is just a constant value, it is still a polynomial function where you just imagine it's multiplied by x to the zero, meaning it is multiplied by one, and hence, why you leave that part out. If you look to the right, this is what a rational function may look like when it is plotted. So again, you have your blue line, and it has kind of some different curves going on, so you'll notice it has kind of a lot of different turning points in the process. Let's review some common properties for rational functions. First, the domain of a rational function is all real numbers except where the denominator h of x equals zero. These points are called undefined points or excluded values where the function f of x is undefined at that particular point. Number two, rational functions are continuous and differentiable for all real x values where the function is defined. So note that this does not work for the undefined values. Number three, vertical asymptotes appear where the denominator h of x equals zero, and the numerator g of x does not equal zero. Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator. And finally, rational functions can sometimes have symmetry depending on how the negative values of the functions reflect. There are many real life applications for polynomial and rational functions. They can be used in economics to model revenue growth or marginal costs. In pharmacology, they can be used to model drug concentration over time. And finally, in engineering, they could be used to model electrical circuits. Polynomial and rational functions both provide a solid base for functions to work with. These functions will often be the simplest types of functions you will work with throughout this course, but note that they are still important to know and understand, even though they tend to be on the simpler side. I recommend taking time on your own to continue exploring polynomial and rational functions, especially if you still feel uncomfortable with what these functions are. Up next, let's explore inverse functions.

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