Chapter 1 | Functions and Graphs
47
d. The function is a parabola with zeros at p =0 and p =25, and it is symmetric about the line p =12.5, so the maximum revenue occurs at a price of p =$12.50 per item. At that price, the revenue is R ( p ) = −1.04(12.5) 2 + 26(12.5) = $162, 500.
Algebraic Functions By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions. Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a rational function is any function of the form f ( x ) = p ( x )/ q ( x ), where p ( x ) and q ( x ) are polynomials. For example, f ( x ) = 3 x −1 5 x +2 and g ( x ) = 4 x 2 +1 are rational functions. A root function is a power function of the form f ( x ) = x 1/ n , where n is a positive integer greater than one. For example, f ( x ) = x 1/2 = x is the square-root function and g ( x ) = x 1/3 = x 3 is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, f ( x ) = 4− x 2 is an algebraic function.
Made with FlippingBook - professional solution for displaying marketing and sales documents online