How do you find the domain and range of #f(x) = sqrt(x+6) /( 6+x)#?

Answer 1

Domain : #x!=-6# from the denominator.
#x>=-6# as an argument for a square root.

Together we get domain #x> -6#

Range: At this point, both the denominator and the numerator are always positive.

So the range is # 0 < f(x) < +oo # graph{1/(sqrt(x+6)) [-10, 10, -5, 5]} Note: You may have noticed that #6+x=x+6=(sqrt(x+6))^2#, so under the given conditions we may rewrite: #f(x)=cancel(sqrt(x+6))/(sqrt(x+6))^cancel2=1/(sqrt(x+6))#
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Answer 2

To find the domain and range of ( f(x) = \frac{\sqrt{x+6}}{6+x} ):

  1. Domain: Set the denominator ( 6+x ) not equal to zero and find the values of ( x ) that satisfy this condition. The domain is all real numbers except ( x = -6 ).

  2. Range: Analyze the behavior of the function as ( x ) approaches positive and negative infinity. Since the function is a rational function with a square root, the range is all real numbers greater than or equal to zero.

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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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