Rules of Exponents

“onto.”The reader should verify that the functional relationship in Figure 2.1 does not have an inverse.

The foregoing discussion suggests that it is not possible to write x = g(y) = f -1(y) until we have determined whether the function y = f(x) is monotonic.Diagrammatically,it is easy to determine whether a function is monotonic by examining its slope.If the slope of the function is positive for all values of x,then the function y = f(x) is monotonically increasing.If the slope of the function is negative for all values of x,then the function y = f(x) is monotonically decreasing.It is easy to see that the linear function y = f(x) = a + bx is a monotonically increasing or decreasing function depending on the value of b.A positive value for b indicates the function is monotonically increasing.A negative value for b indicates that the function is monotonically decreasing.

Because linear functions are monotonic,a one-to-one correspondence must exist between x and y.As a result,all linear functions have a corresponding inverse function.A discussion of the monotonicity of nonlinear functions will be deferred until our discussion of the inverse-function rule in connection with the derivative of a function.

Example. Consider the function

y fx x = () = -2 3 This function is one-to-one because for every value of x there is one,and only one, value for y.When we have solved for x,this function becomes

This function is also one-to-one,since for each value of y there is one,and only one, value for x.Since the original function is both one-to-one and onto,there is a oneto-one correspondence between x and y.Conversely,since the inverse function is also one-to-one and onto,there is a one-to-one correspondence between y and x. Finally,both the original function and the inverse function are monotonically decreasing,since the slope of each function is negative.

x gy f y y = () = () = - ( )-1 23 13

RULES OF EXPONENTS

Before considering nonlinear equations,some students may find it useful to review the rules of exponents.We begin this review by denoting the values of x and y as any two real numbers,and m and n any two positive integers.5 Consider the following rules of dealing with exponents: Rule 1: (2.18) x x xm n m n ◊ =

Examples

a. x2 ◊ x3 = x2+3 = x5 = x ◊ x ◊ x ◊ x ◊ x b.32 ◊ 34 = 32+4 = 36 = 3 ◊ 3 ◊ 3 ◊ 3 ◊ 3 ◊ 3 = 729 c. yr ◊ ys = yr+s

5 A real number is defined as the ratio of any two integers.