How do you find the domain and range of #1/(x+6)#?
1 Answer
Domain:
Range:
Explanation:
The domain is all possible values of
We see that the function is only undefined if the denominator is 0, meaning that
#x+6 = 0# This tells us that
#x# cannot be#-6# .
So we can say our domain is:
(This is just a fancy way of saying "
#x# can be all real numbers except for -6")
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Range is a little harder to find. We need to find all possible values that
Let's think about it this way: what does the graph of
graph{y = 1/(x+6) [-13.71, 6.29, -4.76, 5.24]}
We need to find all possible
When
#x < -6# , we can see that the function#1/(x+6)# will be negative, since#x+6 # will be negative.As we approach
#x = -6# from the left side, the function flies downwards towards#-oo# , hitting every possible negative value.As we approach
#x = -oo# , the function tends towards zero, but never actually reaches it. This is because the denominator is getting bigger and bigger, so the fraction is getting closer and closer to 0 without ever reaching it.Therefore, from
#x = -oo# to#x=-6# , we can say that we will hit all possible negative values of#y# .The same logic can be used for the positive side of the graph.
As we approach
#x = -6# from the right side, the function flies upwards towards#oo# , hitting every possible positive value.As we approach
#x = oo# , the function tends towards zero but never actually reaches it.Therefore, from
#x = -6# to#x = oo# , we can say that we will hit all possible positive values of#y# .
We've checked every possible
#y# is a real number
#y# is positive OR negative
In other words:
#y in RR, y ne 0#
This is the range of
Final Answer
