Domain and range of g(x) = (3x^2+4)/(x-2)

BertonCO5

BertonCO5

Answered question

2022-11-24

Domain and range of g ( x ) = 3 x 2 + 4 x 2
for the domain, I find, { x | x 2 }
and for the range
y = 3 x 2 + 4 x 2 and when x = 2 ϵ, y 3 ( 2 ϵ ) 2 + 4 ( 2 ϵ ) 2 = K ϵ
When x = 2 + ϵ, y 3 ( 2 + ϵ ) 2 + 4 ( 2 + ϵ ) 2 = K ϵ
Therefore, ran ( g ( x ) ) = ( , )
have I done this right or am I missing something? For instance, I thought about the case when x , but this looks like y 1 in this case, nothing remarkable. I don't even know if that is what we say when we have an 2 ....Do we just say that is one? Or do we say that is still?

Answer & Explanation

Karly Donovan

Karly Donovan

Beginner2022-11-25Added 5 answers

Fix y R . Then you look for x such that
3 x 2 + 4 x 2 = y .
Solve this equation: you get
3 x 2 + 4 = x y 2 y 3 x 2 y x + 2 y + 4 = 0 ,
which gives
x = y ± y 2 24 y 48 6 .
So your equation admits real solutions only if y 2 24 y 48 0. But this is y 2 24 y + 144 192 = ( y 12 ) 2 192 0, i.e. | y 12 | 192 . It is clear that this is not true for every y, since for example y = 0 does not satisfy the inequality. So the range is not R , but is given by those points y satisfying | y 12 | 192
One more comment on your last observation. If x then
g ( x ) = x 2 ( 3 + 4 x 2 ) x ( 1 2 x ) = x ( 3 + 4 x 2 ) ( 1 2 x ) 3 1 = .
goorst9Bi

goorst9Bi

Beginner2022-11-26Added 1 answers

Your answer is wrong. For example, 0 is not in the range of the function. Here is a hint for finding the range: consider the range on ( 2 , ) and ( , 2 ). On these intervals the function is continuous so it will take all values between two limits. So find the max and the min on these two intervals.

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