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:

Step1: Factor numerator/denominator

For $f(x)=\frac{x-1}{x^2-1}$:
Numerator: $x-1$
Denominator: $x^2-1=(x-1)(x+1)$
Common factor: $x-1$

Step2: Factor numerator/denominator

For $f(x)=\frac{x^2-9}{x^2+7x+12}$:
Numerator: $x^2-9=(x-3)(x+3)$
Denominator: $x^2+7x+12=(x+3)(x+4)$
Common factor: $x+3$

Step3: Factor numerator/denominator

For $f(x)=\frac{x^2+4x+4}{x^2+2x-8}$:
Numerator: $x^2+4x+4=(x+2)^2$
Denominator: $x^2+2x-8=(x+4)(x-2)$
No common factors

Step4: Factor numerator/denominator

For $f(x)=\frac{x+7}{x^2+5x-14}$:
Numerator: $x+7$
Denominator: $x^2+5x-14=(x+7)(x-2)$
Common factor: $x+7$

Answer:

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