Advanced is Basic
f ( x ) = ln ( x 2 − 5 x − 2 4 − x − 2 )
What is the domain of the definition of the function?
Try this first basic .
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1 solution
ln0 is also not defined hence x should not be -28/9=-3.11111111111 but in the domain this is included so answer should be none Niranjan Khanderia sir
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Did not understand where you got -28/9. But note that when x<0, (-x-2)>0....
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x 2 − 5 x − 2 4 > x + 2 you will get x < − 2 8 / 9
I think the answer is ( − i n f i n i t y , − 2 8 / 9 ]
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@Rindell Mabunga
–
We say
(
−
∞
,
−
3
)
, it includes
(
−
∞
,
−
9
2
9
)
.
(
−
∞
,
−
9
2
9
)
, it DOES NOT include
(
−
∞
,
−
3
)
.
idk why my answer is (-infinity,-28/9).....
If x= -28/9 then f(x)=ln(20/9) not ln0
I have edited the solution for clarity.
Firstly, the square root must be positive, so x 2 − 5 x − 2 4 = ( x − 8 ) ( x + 3 ) ≥ 0 ⟹ x ≤ − 3 o r x ≥ 8 . .
However, if x ≥ 8 , we will show that x 2 − 5 x − 2 4 < x + 2 . Since both sides are positive, we can square both sides to obtain x 2 − 5 x − 2 4 < x 2 + 4 x + 4 , which is obviously true. Since we cannot take ln of a negative number, this is not part of the domain.
If x ≤ − 3 , then we will show that x 2 − 5 x − 2 4 > x + 2 . This is true because the RHS is negative, while the LHS is positive. Hence, we can apply ln to the expression.
So, the domain is ( − ∞ , − 3 ] .