In the text, on page 26, the authors establish their conventions that the domain of the sum or product of two functions is the intersections of the domains of the functions, that is to say the points on the x-axis where both functions are defined. In the case of division of a function f by a function g, there is an additional stipulation that the domain of f/g should not include any points x where g(x) = 0. This means that it is necessary to describe the domain of the sum, product, or quotient, before carrying out any algebraic simplifications.
For example, if f(x) = x for all x and g(x) = x for all x, then the domain of f/g is all x but x = 0 since g(0) = 0. Thus (f/g)(x) = 1 if x is non-zero and (f/g) is not defined at zero.
As another example if f(x) = {sqrt}(x) and g(x) = {sqrt}(x) then the domain of f and the domain of g are both the non-negative real numbers, so the domain of fg cannot contain any negative numbers. Thus (fg)(x) = {sqrt}(x){sqrt}(x) = {sqrt}(x{^2}) = |x| if x > 0 or x = 0, and (fg)(x) is not defined for negative numbers.
In Probem 6, we have (f/g)(x) = (x-1/x-2)/(x+1/x+2). The domain of f is all x other than 2 and the domain of g is all x other than -2. There is one value of x, namely -1, where g(x) = 0 and this also must be excluded from the domain of the quotient (f/g). Thus the domain of f/g is all numbers except for -1, 2, and -2, even though when we simplify algebraically we get (f/g)(x) = ((x-1)(x+2))/((x-2)(x+1)) which appeard to be all right at x = -2.

There are many ways to identify which functions belong to which graphs in the list. Thus we can identify the graph of x/(x{^2} - 9) by looking at the y-intercept and noting which graphs pass through the point (0,0). We can also look for the graphs which are not defined over the points x = 3 and x = -3. Either answer is good, and there can be other answers as well.
Note however that not all answers work. If the equation is cubic, then the graph does not have to have three real roots. It might have one, or three, or even two in case the graph has a local maximum or minimum with a 0 y-coordinate. Compare the graphs of x{^3} - 3x, x{^3} - 3x + 3 and x{^3} - 3x + 2. A similar comment holds about a fifth degree equation. If there are more than three roots, the equation has to have degree greater than three and if there are five roots, the equation has to have degree at least five. But not all fifth degree equations have five roots, for example x{^5} - 1 which has x = 1 as its only root.

More later.

Solution 2 Comments

Thomas Banchoff
Last modified: Thu Sep 19 19:47:49 EDT 2002