Section 1: Introducing Rational Functions

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 7.1 Introducing Rational Functions 603 Version: Fall 2007 7.1 Introducing Rational Functions In the previous chapter, we studied polynomials, functions having equation form p ( x ) = a 0 + a 1 x + a 2 x 2 + · · · + a n x n . (1) Even though this polynomial is presented in ascending powers of x , the leading term of the polynomial is still a n x n , the term with the highest power of x . The degree of the polynomial is the highest power of x present, so in this case, the degree of the polynomial is n . In this section, our study will lead us to the rational functions. Note the root word “ratio” in the term “rational.” Does it remind you of the word “fraction”? It should, as rational functions are functions in a very specific fractional form. Definition 2. A rational function is a function that can be written as a quotient of two polynomial functions. In symbols, the function f ( x ) = a 0 + a 1 x + a 2 x 2 + · · · + a n x n b 0 + b 1 x + b 2 x 2 + · · · + b m x m (3) is called a rational function . For example, f ( x ) = 1 + x x + 2 , g ( x ) = x 2 2 x 3 x + 4 , and h ( x ) = 3 2 x x 2 x 3 + 2 x 2 3 x 5 (4) are rational functions, while f ( x ) = 1 + x x 2 + 1 , g ( x ) = x 2 + 2 x 3 1 + x 1 / 2 3 x 2 , and h ( x ) = x 2 2 x 3 x 2 + 4 x 12 (5) are not rational functions. Each of the functions in equation (4) are rational functions, because in each case, the numerator and denominator of the given expression is a valid polynomial. However, in equation (5) , the numerator of f ( x ) is not a polynomial (polynomials do not allow the square root of the independent variable). Therefore, f is not a rational function. Similarly, the denominator of g ( x ) in equation (5) is not a polynomial. Fractions are not allowed as exponents in polynomials. Thus, g is not a rational function. Finally, in the case of function h in equation (5) , although the radicand (the expression inside the radical) is a rational function, the square root prevents h from being a rational function. Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1
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604 Chapter 7 Rational Functions Version: Fall 2007 An important skill to develop is the ability to draw the graph of a rational function. Let’s begin by drawing the graph of one of the simplest (but most fundamental) rational functions. The Graph of y = 1 /x In all new situations, when we are presented with an equation whose graph we’ve not considered or do not recognize, we begin the process of drawing the graph by creating a table of points that satisfy the equation. It’s important to remember that the graph of an equation is the set of all points that satisfy the equation. We note that zero is not in the domain of y = 1 /x (division by zero makes no sense and is not defined), and create a table of points satisfying the equation shown in Figure 1 . x 10 y 10 x y = 1 /x 3 1 / 3 2 1 / 2 1 1 1 1 2 1 / 2 3 1 / 3 Figure 1. At the right is a table of points satisfying the equation y = 1 /x . These points are plotted as solid dots on the graph at the left.
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