Domain of a function
We want to find the domain of the following function:
[tex]=ln\mleft(^2-6-55\mright)[/tex]
This means that we want to find the x-values that it can take.
STEP 1: analyzing the simplies form of the function
Let's analyze the simpliest form of the function:
[tex]=ln(x)[/tex]
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
STEP 2: domain of the given function
Based on the above we can deduce that for the ln(x) function, what is inside the parenthesis should be higher than 0 on this kind of functions.
This is that for
[tex]=ln\mleft(^2-6-55\mright)[/tex]
then
[tex]^2-6-55>0[/tex]
STEP 3: finding the x values that make x²-6x-55>0 (factoring)
In order to find the values of x that make
[tex]^2-6-55>0[/tex]
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:
[tex]^2-6-55=(x+5)(x-11)[/tex]
Then,
[tex](x+5)(x-11)>0[/tex]
STEP 4: finding the x values that make (x+5)(x-11)>0 (factoring)
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:
[tex](x+5)(x-11)[/tex]
We use the diagram to analyze the sign of their product:
Then
[tex](x+5)(x-11)>0[/tex]
when x < -5 and when x > 11. This is the domain.
Therefore, expressed in set notation:
domain = {x|x∈(-∞, -5)∪(11, ∞)}
Answer: domain = {x | x ∈ (-∞, -5)∪(11, ∞)}
