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Subject Test Math 1

Question 18 of 32

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SAT Subject Test

If f left parenthesis x right parenthesis equals x plus 3 and g left parenthesis x right parenthesis equals fraction numerator x to the power of 2 end exponent minus 9 over denominator x minus 3 end fraction invisible times, which of the following statements are true about the graphs of f and g in the x y minus text plane end text?

I. The graphs are exactly the same.
II. The graphs are the same except when x equals 3.
III. The graphs have an infinite number of points in common.

Select an Answer

I only

Correct Answer: 
No

II only

Correct Answer: 
No

III only

Correct Answer: 
No

I and III

Correct Answer: 
No

II and III

Correct Answer: 
Yes

The correct answer is E.

Statement I is false: The function f has the real line as its domain. The function g has as its domain all values of x except 3, because the value of the expression fraction numerator x to the power of 2 end exponent minus 9 over denominator x minus 3 end fraction invisible times is undefined when 3 is substituted for x. This means that the graph of f contains a point with x-coordinate equal to 3,but the graph of g contains no such point.

Statement II is true: For any x not equal to 3, the expressions x plus 3 and fraction numerator x to the power of 2 end exponent minus 9 over denominator x minus 3 end fraction invisible times give the same value, since for x not equal to 3, the expression fraction numerator x to the power of 2 end exponent minus 9 over denominator x minus 3 end fraction invisible times equals fraction numerator left parenthesis x plus 3 right parenthesis left parenthesis x minus 3 right parenthesis over denominator x minus 3 end fraction equals x plus 3. It follows that for x not equal to 3, the graphs of f and g contain the same point with that x minus text coordinate end text.

Statement III is true: Since statement II is true, for every number x not equal to 3, the point left parenthesis x comma x plus 3 right parenthesis is on both the graph of f and the graph of g. Since there are infinitely many numbers x not equal to 3, the graphs have an infinite number of points in common.

Question Difficulty: 
medium