Domain of a function
We want to find the domain of the following function:
![=ln\mleft(^2-6-55\mright)](https://tex.z-dn.net/?f=%3Dln%5Cmleft%28%5E2-6-55%5Cmright%29)
This means that we want to find the x-values that it can take.
<h2>STEP 1: analyzing the simplies form of the function</h2>
Let's analyze the simpliest form of the function:
![=ln(x)](https://tex.z-dn.net/?f=%3Dln%28x%29)
Its graph is:
Then, for the simpliest form of the function, the x-values can only be higher than 0.
This means that its domain is
domain = x > 0
<h2>STEP 2: domain of the given function</h2>
Based on the above we can deduce that for the <em>ln(x)</em> function, what is inside the parenthesis should be higher than 0 on this kind of functions.
This is that for
![=ln\mleft(^2-6-55\mright)](https://tex.z-dn.net/?f=%3Dln%5Cmleft%28%5E2-6-55%5Cmright%29)
then
![^2-6-55>0](https://tex.z-dn.net/?f=%5E2-6-55%3E0)
<h2>STEP 3: finding the x values that make x²-6x-55>0 (factoring)</h2>
In order to find the values of x that make
![^2-6-55>0](https://tex.z-dn.net/?f=%5E2-6-55%3E0)
we must factor it.
We want to find a pair of numbers that when multiplied give the last term (-55) and when added together give the second term (-6).
For the last term of the polynomial: -55, we have that
(-5) · 11 = 55
5 · (-11) = 11
If we add them:
-5 + 11 = 6
5 - 11 = -6
The pair of numbers that when multiplied give the last term (-55) and when added together give the second term (-6), are: 5 and -11
We use them to factor the polynomial:
![^2-6-55=(x+5)(x-11)](https://tex.z-dn.net/?f=%5E2-6-55%3D%28x%2B5%29%28x-11%29)
Then,
![(x+5)(x-11)>0](https://tex.z-dn.net/?f=%28x%2B5%29%28x-11%29%3E0)
<h2>STEP 4: finding the x values that make (x+5)(x-11)>0 (factoring)</h2>
In order to find them, we are going to separate the factors (x+5) and (x-11) and analyze when they are positive or negative:
Combining them:
Since we are going to multiply both factors:
![(x+5)(x-11)](https://tex.z-dn.net/?f=%28x%2B5%29%28x-11%29)
We use the diagram to analyze the sign of their product:
Then
![(x+5)(x-11)>0](https://tex.z-dn.net/?f=%28x%2B5%29%28x-11%29%3E0)
when x < -5 and when x > 11. This is the domain.
Therefore, expressed in set notation:
domain = {x|x∈(-∞, -5)∪(11, ∞)}
<h2>
Answer: domain = {x | x ∈ (-∞, -5)∪(11, ∞)}</h2>