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How do you find the domain and range of #y=1/(x-4)#?

Answer 1

The only restriction on the domain is #x!=4#
As this would make the numerator #=0#

As #x# nears #4# from above, #y# will be larger and larger, or in "the language": #lim_(x->4+) y = oo# Something like that goes if #x# nears #4# from below: #lim_(x->4-) y = -oo# #x=4# is called the vertical asymptote.

#y# can never reach the value of #0# ( horizontal asymptote), so th range is #y!=0#, or: #lim_(x->oo) y=0#

graph{1/(x-4) [-14.96, 5.76, -5.04, -4.24]}

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Answer 2

Eli Assouline

To find the domain of ( y = \frac{1}{x-4} ), we need to identify any values of ( x ) that would make the denominator zero, since division by zero is undefined. So, the domain is all real numbers except ( x = 4 ).

For the range, we need to consider what happens to the function as ( x ) approaches positive and negative infinity. As ( x ) approaches positive infinity, ( y ) approaches zero. As ( x ) approaches negative infinity, ( y ) also approaches zero. Therefore, the range of the function is all real numbers except zero.

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Answer 3

Abigail Allen

The domain of ( y = \frac{1}{x - 4} ) is all real numbers except ( x = 4 ). The range of the function is all real numbers except ( y = 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.