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Subject Test Math 2

Question 2 of 28

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SAT Subject Test

If  f left parenthesis x right parenthesis equals fraction numerator 3 x plus 12 over denominator 2 x minus 12 end fraction, what value does f invisible times left parenthesis x right parenthesis approach as x gets infinitely larger?

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negative 6

Correct Answer: 
No

negative 3 over 2

Correct Answer: 
No

negative 1

Correct Answer: 
No

2 over 3

Correct Answer: 
No

3 over 2

Correct Answer: 
Yes

The correct answer is E.

One way to determine the value that f invisible times left parenthesis x right parenthesis approaches as x gets infinitely larger is to rewrite the definition of the function to use only negative powers of x and then reason about the behavior of negative powers of x as x gets infinitely larger. Since the question is only concerned with what happens to fraction numerator 3 x plus 12 over denominator 2 x minus 12 end fraction as x gets infinitely larger, one can assume that x is positive. For x not equal to 0, the expression fraction numerator 3 x plus 12 over denominator 2 x minus 12 end fraction is equivalent to the expression fraction numerator fraction numerator 1 over denominator x end fraction left parenthesis 3 x plus 12 right parenthesis over denominator fraction numerator 1 over denominator x end fraction left parenthesis 2 x minus 12 right parenthesis end fraction equals fraction numerator 3 plus fraction numerator 12 over denominator x end fraction over denominator 2 minus fraction numerator 12 over denominator x end fraction end fraction. As x gets infinitely larger, the expression fraction numerator 12 over denominator x end fraction approaches the value 0, so as x gets infinitely larger, the expression fraction numerator 3 plus fraction numerator 12 over denominator x end fraction over denominator 2 minus fraction numerator 12 over denominator x end fraction end fraction approaches the value fraction numerator 3 plus 0 over denominator 2 minus 0 end fraction equals fraction numerator 3 over denominator 2 end fraction. Thus, as x gets infinitely larger, f invisible times left parenthesis x right parenthesis approaches fraction numerator 3 over denominator 2 end fraction.

Alternatively, one can use a graphing calculator to estimate the height of the horizontal asymptote for the function f left parenthesis x right parenthesis equals fraction numerator 3 x plus 12 over denominator 2 x minus 12 end fraction. Graph the function y equals fraction numerator 3 x plus 12 over denominator 2 x minus 12 end fraction on an interval with “large” x minus text values end text, say, from x equals 100 to x equals 1 , 000.

The figure shows a graph in the x y plane. The integers 100 to 1,000, in increments of 100, are indicated on the horizontal axis, and the numbers from 0 to 2 point 5, in increments of 0 point 5, are indicated on the vertical axis. A curve starts at x equal to 100 and y equal to 1 point 7, moves down very slightly to y equal to 1 point 5 at x equal to 200, and then extends almost horizontally to the right.

By examining the graph, the y minus text values end text all seem very close to 1.5. Graph the function again, from, say, x equals 1 , 000 to x equals 10 , 000.

The figure shows a graph in the x y plane. The integers 1,000 to 10,000, in increments of 1,000, are indicated on the horizontal axis, and the numbers from 0 to 2 point 5, in increments of 0 point 5, are indicated on the vertical axis. A line starts at x equal to 1,000, y equal to 1 point 5, and extends horizontally to the right.

The y minus text values end text vary even less from 1.5. In fact, to the scale of the coordinate plane shown, the graph of the function f left parenthesis x right parenthesis equals fraction numerator 3 x plus 12 over denominator 2 x minus 12 end fraction is nearly indistinguishable from the asymptotic line y equals 1.5. This suggests that as x gets infinitely larger, f invisible times left parenthesis x right parenthesis approaches 1.5, that is, fraction numerator 3 over denominator 2 end fraction
Note: The algebraic method is preferable, as it provides a proof that guarantees that the value f invisible times left parenthesis x right parenthesis approaches is fraction numerator 3 over denominator 2 end fraction. Although the graphical method worked in this case, it does not provide a complete justification; for example, the graphical method does not ensure that the graph resembles a horizontal line for “very large” x minus text values end text such as 10 to the power of 100 end exponent less or equal than x less or equal than 10 to the power of 101 end exponent.

Question Difficulty: 
easy