Question

Describe where the function has a hole and how you found your answer.

Answer 1

Step 1:

Write the function

`f(x)\text{ = }\frac{x^2+\text{ 7x + 10}}{x^2\text{ + 9x + 20}}\text{ }`

Step 2:

Factorize both the numerator and the denominator.

`\begin{gathered} f(x)\text{ = }\frac{x^2\text{ + 2x + 5x + 10}}{x^2\text{ + 4x + 5x + 20}} \\ f(x)\text{ = }\frac{x(x\text{ + 2) + 5(x + 2)}}{x(x\text{ + 4) + 5 (x + 4)}} \\ f(x)\text{ = }\frac{(x\text{ + 5)(x +2)}}{(x\text{ + 5)(x + 4)}} \end{gathered}`

Step 3:

A hole is a common factor between the numerator and the denominator.

Hole: x + 5 = 0

x = -5

Final answer

Hole is -5