Difference between revisions of "1995 AHSME Problems/Problem 12"
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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 <math>f(5) = 5</math> tells us that the value of the constant is <math>5</math>, and thus that <math>f(x) = 5</math>. This leads to <math>f(0)=5\Rightarrow \mathrm{(D)}</math> | 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 <math>f(5) = 5</math> tells us that the value of the constant is <math>5</math>, and thus that <math>f(x) = 5</math>. This leads to <math>f(0)=5\Rightarrow \mathrm{(D)}</math> | ||
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+ | ==Solution 3 (== | ||
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+ | It should be very clear that <math>f(1)<f(2)</math> and <math>f(3)>f(4)</math> is contradictory because of the fact that linear functions are monotonic. The only thing that makes sense is <math>f(1)=f(2)</math>, and <math>f(3)=f(4)</math>. This means that <math>f(x)</math> has a slope of 0<math>. So </math>f(x)=5<math>. So </math>f(0)=5<math>. Select </math>\boxed{D}$. | ||
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+ | ~hastapasta | ||
==See also== | ==See also== |
Revision as of 12:25, 4 May 2022
Contents
[hide]Problem
Let be a linear function with the properties that and . Which of the following is true?
Solution 1
A linear function has the property that either for all , or for all . Since , . Since , . And if for , then is a constant function. Since ,
Solution 2
If is a linear function, the statement states that the slope of the line is nonnegative: it is either positive or zero.
Similarly, the statement states that the slope of the line is nonpositive: it is either negative or zero.
Since the slope of a linear function can only have one value, it must be zero, and thus the function is a constant. The statement tells us that the value of the constant is , and thus that . This leads to
Solution 3 (
It should be very clear that and is contradictory because of the fact that linear functions are monotonic. The only thing that makes sense is , and . This means that has a slope of 0f(x)=5f(0)=5\boxed{D}$.
~hastapasta
See also
1995 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.