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Given that f(x) = x^k where k < 0, what is the range of f(x) on the interval [1, ∞)? Please explain, I want to understand how the answer is gotten. Thank you

 Dec 7, 2019
 #1
avatar+108732 
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Given that f(x) = x^k where k < 0, what is the range of f(x) on the interval [1, ∞)? Please explain, I want to understand how the answer is gotten. 

 

Ok, I will try to explain. 

 

\( f(x) = x^k \;\;\;where\;\;\; k < 0, \;\;\:and \;\;\;\ x\ge1\\ let\;\; n=-k \;\;\;so\;\;\;n>0\\ f(x) = x^{-n}=\frac{1}{x^n}\\ now\;\; x\ge1 \;\; and\;\; n>0\;\;\;\\so\\ x^n\ge1\\ so\\ 0< \frac{1}{x^n} \le1\\ 0< x^{-n}\le1\\ 0< x^{k}\le1\\ \)

 

Here is the graph

https://www.desmos.com/calculator/y0xfckw4zn

 Dec 7, 2019
 #2
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Thank you so much, Melody. I understand this problem.

 Dec 8, 2019
 #3
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You are very welcome EpicWaters.

Melody  Dec 8, 2019

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