How do you find the domain and range of #f(x) = (3x + 1)/ (sqrt(x^2 + x - 2) ) #?

Answer 1

The domain is #x in (-oo,-2) uu (1,oo)#
The range is #y in (-oo,-sqrt80/3) uu(sqrt80/3, oo)#

The denominator must be #!=0#

Consequently,

#sqrt(x^2+x-2)!=0#, #=>#, #x^2+x-2>0#
#(x+2)(x-1)>0#
As the coefficient of #x^2# is #>0#, so
#x in (-oo,-2) uu (1,oo)#
The domain is #x in (-oo,-2) uu (1,oo)#

To determine the range, follow these steps:

Let #y=(3x+1)/sqrt(x^2+x-2)#

Changing the order of this equation

#ysqrt(x^2+x-2)=(3x+1)#

balancing both sides

#(ysqrt(x^2+x-2))^2=(3x+1)^2#
#y^2(x^2+x-2)=9x^2+6x+1#

Moving things around

#x^2(y^2-9)+x(y^2-6)-(2y^2+1)=0#
This is a quadratic equation in #x#, in order to have solutions, the discriminant must be #>=0#

The person who discriminates is

#Delta=b^2-4ac=(y^2-6)^2+4(y^2-9)(2y^2+1)>=0#
#y^4-12y^2+36+8y^4-68y^2-36>=0#
#9y^4-80y^2>=0#
#y^2(9y^2-80)>=0#
#y=0#, #S=O/#
#y=+-sqrt80/3#
The range is #y in (-oo,-sqrt80/3) uu(sqrt80/3, oo)#

graph{sqrt(x^2+x-2) [-28.87, 28.88, -14.43, 14.43]}/(3x+1)

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Answer 2

To find the domain of ( f(x) = \frac{3x + 1}{\sqrt{x^2 + x - 2}} ), we need to ensure that the denominator is not zero and that the expression inside the square root is non-negative. The domain is all real numbers except where the denominator is zero and where the expression inside the square root becomes negative.

To find the range, we first need to analyze the behavior of the function as ( x ) approaches positive and negative infinity. Then, we look for any horizontal asymptotes or restrictions on the function's values due to its behavior.

We find that the range of the function is all real numbers except for those values where the denominator approaches zero or where the function is undefined due to the square root of a negative number.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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