What is the domain and range of #g(x) =(5x)/(x^2-36)#?

Answer 1

#x inRR,x!=+-6#
#y inRR,y!=0#

The denominator of g(x) cannot be zero as this would make g(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be.

#"solve "x^2-36=0rArr(x-6)(x+6)=0#
#rArrx=+-6larrcolor(red)" are excluded values"#
#rArr"domain is "x inRR,x!=+-6#
#"or in interval notation as"#
#(-oo,-6)uu(-6,6)uu(6,+oo)#
#"for range divide terms on numerator/denominator by the"# #"highest power of x that is "x^2#
#g(x)=((5x)/x^2)/(x^2/x^2-36/x^2)=(5/x)/(1-36/x^2)#
#"as "xto+-oo,g(x)to0/(1-0)#
#rArry=0larrcolor(red)"is an excluded value"#
#rArr"range is "y inRR,y!=0#
#(-oo,0)uu(0,+oo)larrcolor(blue)"in interval notation"# graph{(5x)/(x^2-36) [-10, 10, -5, 5]}
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Answer 2

The domain of the function (g(x) = \frac{5x}{x^2 - 36}) consists of all real numbers except those that make the denominator equal to zero. Thus, the domain is all real numbers except (x = 6) and (x = -6), since those values would result in division by zero.

The range of the function depends on the behavior of the function as (x) approaches positive and negative infinity. As (x) approaches positive or negative infinity, the function approaches zero. Therefore, the range of the function is all real numbers except (y = 0).

In summary:

  • Domain: (x \in \mathbb{R}) such that (x \neq 6) and (x \neq -6).
  • Range: (y \in \mathbb{R}) such that (y \neq 0).
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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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