Question

True or False The graph of a rational function sometimes has a hole.

Short answer

True

Step-by-step solution

Understanding Rational Functions

Rational functions are functions of the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) is not zero.

Identifying Potential Holes

A hole in a rational function occurs when both the numerator \( P(x) \) and the denominator \( Q(x) \) have a common factor that can be cancelled, leading to an undefined point in the function.

Analyzing the Condition for Holes

For a rational function \( \frac{P(x)}{Q(x)} \), if there exists a \( x = a \) that makes both \( P(a) = 0 \) and \( Q(a) = 0 \), after canceling the common factor, the function is undefined at \( x = a \), resulting in a hole at that point.

Determining the General Case

Since not all rational functions may have factors in the numerator and the denominator that can be cancelled out, not every rational function will have a hole. This means holes occur in some, but not all, rational functions.

Conclusion

Given that holes occur only when the conditions for a common factor are met, it is true that the graph of a rational function sometimes has a hole.

Key concepts

holes in rational functions

Rational functions sometimes have holes in their graphs. These holes occur where there are certain specific conditions. Have you ever seen a graph where it just skips a point? That's due to a 'hole'. A hole happens when both the numerator and the denominator of the fraction have a common factor that can be cancelled.

For example, consider the function: \( \frac{(x-2)(x+3)}{(x-2)(x-4)} \). Here, both the numerator and the denominator have the term \( (x-2) \).

If we cancel out \( (x-2) \), it is removed from the function, but at \( x=2 \), the original function is undefined, creating a hole.

This type of hole is not always present in every rational function, making it a special case that only occurs under specific conditions. Most importantly, the hole comes from these common factors.

common factors in polynomials

When working with rational functions, you might come across common factors in the polynomials of both the numerator and the denominator. These factors are key to understanding holes.

Remember, a rational function generally looks like this: \( \frac{P(x)}{Q(x)} \). If there is a common factor in both \( P(x) \) and \( Q(x) \), you can simplify the function by canceling out these factors.

Here's an example:
Imagine we have the rational function: \( \frac{x(x-1)}{(x-1)(x+2)} \). The factor \( (x-1) \) appears in both the numerator and the denominator. By canceling \( (x-1) \), the simplified form becomes \( \frac{x}{x+2} \).

Even though we simplified the expression, the point \( x = 1 \) creates a hole in the function because that value would make the denominator zero before simplification.

The trick is to recognize that these common factors point directly to where potential holes can occur. Always keep an eye on common factors when dealing with rational functions.

undefined points in functions

In rational functions, certain points can make the denominator equal zero. These points are considered undefined because you cannot divide by zero.

Take this rational function as an example: \( \frac{1}{x-3} \). Here, the function becomes undefined at \( x = 3 \) because the denominator would be zero.

For more complex rational functions involving polynomials, you'll often need to factor the numerator and denominator to identify these undefined points.

Let's consider: \( \frac{x^2 - 4}{x^2 - 5x + 6} \). When you factor these, it becomes: \( \frac{(x-2)(x+2)}{(x-2)(x-3)} \).

After canceling the common factor \( (x-2) \), you're left with \( \frac{x+2}{x-3} \), but don't forget there's still an undefined point at \( x = 2 \) from the original form.

Undefined points mean the function does not exist for those values, leading to either vertical asymptotes or holes depending on whether a common factor was canceled during simplification.