Question What are the coordinates of the hole, if one exists. (-1, -2) (-1, ) there isn't one (-1, )
Studdy Solution
STEP 1
What is this asking? We're on the hunt for a sneaky "hole" in this function, which is basically a point where the function looks like it's continuous, but it's actually undefined! Watch out! Don't just simplify and forget to check for values of that make the original function undefined!
STEP 2
1. Factor the numerator and denominator
2. Identify and simplify common factors
3. Find the x-coordinate of the hole
4. Find the y-coordinate of the hole
STEP 3
Let's **factor** the numerator .
We can pull out a from both terms, giving us .
See how neat that is?
STEP 4
Now, let's **factor** the denominator .
We're looking for two numbers that multiply to -3 and add up to -2.
Those numbers are -3 and 1!
So, the factored form is .
STEP 5
Now, look at our factored function: We have an in both the numerator and the denominator!
Since when , we can simplify the function to But remember, this simplification is only valid when is *not* equal to **-1**, because that would make the original denominator zero, and we can't divide by zero!
STEP 6
That value of where we had to be careful, , is the x-coordinate of our hole!
STEP 7
To find the y-coordinate, we plug our x-coordinate of the hole, , into the simplified version of our function:
STEP 8
Let's calculate this: Dividing both the numerator and denominator by -2 gives us: So, the y-coordinate of our hole is !
STEP 9
The coordinates of the hole are .
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