Question

What is the coordinates for the location of the hole?

Answer 1

(1, -3)

Step-by-step explanation:

The function contains a hole if there is a value of x that makes the numerator and denominator equal to 0.

The values that make the numerator equal to 0 can be calculated solving the following equation

x² + x - 2 = 0

(x + 2)(x - 1) = 0

Then

x + 2 = 0

x + 2 - 2 = 0 - 2

x = -2

or

x - 1 = 0

x - 1 + 1 = 0 + 1

x = 1

So, the values are x = -2 and x = 1

In the same way, the values that make the denominator equal to 0 are

x² - 3x + 2 = 0

(x - 2)(x - 1) = 0

Then

x - 2 = 0

x - 2 + 2 = 0 + 2

x = 2

or

x - 1 = 0

x - 1 + 1 = 0 + 1

x = 1

So, the values are x = 2 and x = 1.

It means that there is a hole in x = 1. To know the coordinates for the location of the hole, we need to simplify the initial expression as:

`\begin{gathered} f(x)=(x^2+x-2)/(x^2-3x+2) \\ f(x)=((x+2)(x-1))/((x-2)(x-1)) \\ f(x)=((x+2))/((x-2)) \end{gathered}`

Now, we can replace x by 1 to get:

`f(1)=(1+2)/(1-2)=(3)/(-1)=-3`

So, the coordinates for the location of the hole are (1, -3)