Question 1117852: Look at this function: 𝑓(𝑥) = ln(𝑥^2 + 2𝑥 − 2).
1.) What is the domain?
2.) What are the equations of all asymptotes of 𝑓(𝑥).
3.) Find the solution set (solve) for the inequality 𝑓(𝑥) ≤ 0, write the solution set in interval notation.
Found 2 solutions by solver91311, Boreal:
Answer by solver91311(24713)   (Show Source):
Answer by Boreal(15235)  (Show Source):
You can put this solution on YOUR website! Look first at the graph of x^2+2x-2
The roots are done using the quadratic formula
x=(1/2){-2+/- sqrt(4+8)}
This is -1+/-sqrt(3) or -1+sqrt(3) or x > 0.732.
This is -1-sqrt(3) or x < -1.732
When x is between those values, the ln does not exist
The domain is therefore (-oo, -2.732) and (0.732, oo)
From the left side, as x approaches -2.732 from the negative, or to 0.732 from the right, the ln becomes closer to 0 and the ln becomes large negative.
The asymptotes are x=-1-sqrt(3) and x= -1+sqrt (3)
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For the function being <0, one needs values of the quadratic less than 1, since ln (1) is 0, and greater than 0, since ln 0 doesn't exist.
x^2+2x-2=1
x^2+2x-3=0
(x+3)(x-1)=0
x=1, -3
So (-3, -1-sqrt(3)) and (-1+sqrt(3), 1) the function is < 0
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