Problem 35 - Calculus of a Single Variable

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The zeros of \(f(x)=\frac{p(x)}{q(x)}\) coincide with the zeros of \(p(x)\)

Step-by-Step Solution

Answer

The statement is false. A counterexample is when \(p(x)\) and \(q(x)\) are both zero at the same x-value, thus \(f(x)= \frac{p(x)}{q(x)}\) is undefined, not zero. So the zeros of \(f(x)= \frac{p(x)}{q(x)}\) do not always coincide with the zeros of \(p(x)\).

Step 1: Recognition and Identification

Recognize that the zeros of a function are the x-values at which the function equals zero. For a rational function such as \(f(x)= \frac{p(x)}{q(x)}\), it's clear that when \(p(x)\), the numerator, is zero, the whole function \(f(x)\) will be zero, regardless of \(q(x)\), the denominator.

Step 2: Reasoning and Elimination

Interestingly, for nonzero \(q(x)\), when \(p(x)=0\), \(f(x)\) will indeed be 0. But it's crucial to remember that if \(q(x)=0\) at the same x-value where \(p(x)\) is zero, the function \(f(x)\) would be undefined rather than zero, contradicting the original claim.

Step 3: Counterexample

To demonstrate this falsehood, look at an example where \(p(x) = x\) and \(q(x) = x\). For \(x = 0\), both \(p(x)\) and \(q(x)\) are zero, thus making \(f(x)\) undefined, not zero, representing a contradiction to the original statement.