Chapter A.3, Problem 32E

Solution

To explain why the restriction $a\ne b$ is needed in the reduced form of the expression $\frac{\frac{1}{a^2}-\frac{1}{b^2}}{\frac{1}{a}-\frac{1}{b}}$ .

Given:

The reduced form of the given expression is $\frac{a+b}{ab}$ .

Concept Used:

• A domain is the set of all possible input values for a function.
• To find the domain of a function, set the function equal to y and solve for x . This will give you the set of all x -values for which the function produces a real y -value.
• If a factor is common in the numerator and denominator, then the corresponding zero of that factor is excluded from the domain of the original expression.

Calculation:

Note that if $a=b$ , the original expression is undefined as the denominator becomes 0. So, the original expression as well as the reduced form must have the restriction $a\ne b$ .

Since the original expression is undefined for $a=b$ , the restriction $a\ne b$ is needed on the reduced form.