Question

which functions have removable discontinuities (holes)? check all of the boxes that apply.

\\(f(x) = \frac{x - 1}{x^2 - 1}\\)

\\(f(x) = \frac{x^2 - 9}{x^2 + 7x + 12}\\)

\\(f(x) = \frac{x^2 + 4x + 4}{x^2 + 2x - 8}\\)

\\(f(x) = \frac{x + 7}{x^2 + 5x - 14}\\)

Explanation:

🆕 New Concept Discovered: Removable Discontinuities (Holes)
A hole occurs when a factor cancels out from both the numerator and denominator.

Step 1: Analyze the first function

Factor the numerator and denominator of \( f(x) = \frac{x - 1}{x^2 - 1} \):

$$ f(x) = \frac{x - 1}{(x - 1)(x + 1)} $$

The factor \( (x - 1) \) is common to both the numerator and denominator. Canceling this factor leaves a hole at \( x = 1 \).

This function has a removable discontinuity.

Step 2: Analyze the second function

Factor the numerator and denominator of \( f(x) = \frac{x^2 - 9}{x^2 + 7x + 12} \):

$$ f(x) = \frac{(x - 3)(x + 3)}{(x + 3)(x + 4)} $$

The factor \( (x + 3) \) is common to both the numerator and denominator. Canceling this factor leaves a hole at \( x = -3 \).

This function has a removable discontinuity.

Step 3: Analyze the third function

Factor the numerator and denominator of \( f(x) = \frac{x^2 + 4x + 4}{x^2 + 2x - 8} \):

$$ f(x) = \frac{(x + 2)(x + 2)}{(x + 4)(x - 2)} $$

There are no common factors that can be canceled between the numerator and the denominator. The discontinuities at \( x = -4 \) and \( x = 2 \) are vertical asymptotes (non-removable).

This function does not have a removable discontinuity.

Step 4: Analyze the fourth function

Factor the numerator and denominator of \( f(x) = \frac{x + 7}{x^2 + 5x - 14} \):

$$ f(x) = \frac{x + 7}{(x + 7)(x - 2)} $$

The factor \( (x + 7) \) is common to both the numerator and denominator. Canceling this factor leaves a hole at \( x = -7 \).

This function has a removable discontinuity.

Answer:

The functions that have removable discontinuities (holes) are:

• \( f(x) = \frac{x - 1}{x^2 - 1} \)
• \( f(x) = \frac{x^2 - 9}{x^2 + 7x + 12} \)
• \( f(x) = \frac{x + 7}{x^2 + 5x - 14} \)