1995 AHSME Problems/Problem 12
Problem
Let be a linear function with the properties that and . Which of the following is true?
Solution 1
A linear function has the property that either for all , or for all . Since , . Since , . And if for , then is a constant function. Since ,
Solution 2
If is a linear function, the statement states that the slope of the line is nonnegative: it is either positive or zero.
Similarly, the statement states that the slope of the line is nonpositive: it is either negative or zero.
Since the slope of a linear function can only have one value, it must be zero, and thus the function is a constant. The statement tells us that the value of the constant is , and thus that . This leads to
Solution 3 (Common Sense)
It should be very clear that and is contradictory because of the fact that linear functions are monotonic. The only thing that makes sense is , and . This means that has a slope of . So . So . Select .
~hastapasta
See also
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Followed by Problem 13 | |
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