Englishfor English speakers
analytically
Adverb
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When someone studies something analytically, that person studies it in an analytical manner.
independent
Adjective
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If something or someone is independent, they are free; they are not dependent on anything.
independent
Noun
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An independent is a person who does not relies on another person for support.
The USA declared itself independent from Britain in 1776.
function
Noun
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Something or someone's function is its purpose or its job.
The army sometimes performs a policing function.
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A function is an organised event.
I met Sylvia at a church function last week.
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If mathX/math is a function of mathY/math, then mathX/math exists because of or is affected by mathY/math
The speed of the computer is a function of its main processor.
When you see, mathY=f(X)/math, you read it, "mathY/math is a function of mathX/math". Here the value of mathY/math depends on the value assigned to the variable mathX/math.
function
Verb
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If something is functioning, it is working and is not completely broken.
Her heart stopped, but her brain continued to function.
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If something functions as mathX/math, it does the job of mathX/math.
In this sentence, "sentence" functions as the subject.