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Subject Test Math 2

Question 15 of 28

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SAT Subject Test

What is the range of the function defined by f left parenthesis x right parenthesis equals fraction numerator 1 over denominator x end fraction plus 2?

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All real numbers

Correct Answer: 
No

All real numbers except negative 1 half

Correct Answer: 
No

All real numbers except 0

Correct Answer: 
No

All real numbers except 2

Correct Answer: 
Yes

All real numbers between 2 and 3

Correct Answer: 
No

The correct answer is D.

The range of the function defined by f left parenthesis x right parenthesis equals fraction numerator 1 over denominator x end fraction plus 2 is the set of y minus text values end text such that y equals fraction numerator 1 over denominator x end fraction plus 2 for some x minus text value end text.

One way to determine the range of the function defined by f left parenthesis x right parenthesis equals fraction numerator 1 over denominator x end fraction plus 2 is to solve the equation y equals fraction numerator 1 over denominator x end fraction plus 2 for x and then determine which y minus text values end text correspond to at least one x minus text value end text. To solve y equals fraction numerator 1 over denominator x end fraction plus 2 for x, first subtract 2 from both sides to get y minus 2 equals fraction numerator 1 over denominator x end fraction and then take the reciprocal of both sides to get x equals fraction numerator 1 over denominator y minus 2 end fraction. The equation x equals fraction numerator 1 over denominator y minus 2 end fraction shows that for any y minus text value end text other than 2, there is an x minus text value end text such that y equals fraction numerator 1 over denominator x end fraction plus 2, and that there is no such x minus text value end text for y equals 2. Therefore, the range of the function defined by f left parenthesis x right parenthesis equals fraction numerator 1 over denominator x end fraction plus 2 is all real numbers except 2.

Alternatively, one can reason about the possible values of the term fraction numerator 1 over denominator x end fraction. The expression fraction numerator 1 over denominator x end fraction can take on any value except 0, so the expression fraction numerator 1 over denominator x end fraction plus 2 can take on any value except 2. Therefore, the range of the function defined by f left parenthesis x right parenthesis equals fraction numerator 1 over denominator x end fraction plus 2 is all real numbers except 2.

Question Difficulty: 
medium