Difference between revisions of "1975 AHSME Problems/Problem 21"
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m (Small fix to IV Latex $IV: f(b) > f(a) if b > a$ to $IV: f(b) > f(a) \text{ if } b > a$) |
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− | + | ==Problem== | |
Suppose <math>f(x)</math> is defined for all real numbers <math>x; f(x) > 0</math> for all <math>x;</math> and <math>f(a)f(b) = f(a + b)</math> for all <math>a</math> and <math>b</math>. Which of the following statements are true? | Suppose <math>f(x)</math> is defined for all real numbers <math>x; f(x) > 0</math> for all <math>x;</math> and <math>f(a)f(b) = f(a + b)</math> for all <math>a</math> and <math>b</math>. Which of the following statements are true? | ||
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==Solution== | ==Solution== | ||
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+ | <math>I: f(0) = 1</math> | ||
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+ | Let <math>b = 0</math>. Our equation becomes <math>f(a)f(0) = f(a)</math>, so <math>f(0) = 1</math>. Therefore <math>I</math> is always true. | ||
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+ | <math>II: f(-a) = \frac{1}{f(a)} \text{ for all } a</math> | ||
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+ | Let <math>b = -a</math>. Our equation becomes <math>f(a)f(-a) = f(0) = 1 \longrightarrow f(-a) = \frac{1}{f(a)}</math>. Therefore <math>II</math> is always true. | ||
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+ | <math>III: f(a) = \sqrt[3] {f(3a)} \text{ for all } a</math> | ||
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+ | First let <math>b = a</math>. We get <math>f(a)f(a) = f(2a)</math>. Now let <math>b = 2a</math>, giving us <math>f(3a) = f(a)f(2a) = f(a)^3 \longrightarrow f(a) = \sqrt[3] {f(3a)}</math>. Therefore <math>III</math> is always true. | ||
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+ | <math>IV: f(b) > f(a) \text{ if } b > a</math> | ||
+ | This is false. Let <math>f(x) = 2^{-x}</math>, for example. It satisfies the conditions but makes <math>IV</math> false. Therefore <math>IV</math> is not always true. | ||
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+ | Since <math>I, II, \text{ and } III</math> are true, the answer is <math>\boxed{\textbf{(D)}}</math>. | ||
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+ | - mako17 | ||
==See Also== | ==See Also== | ||
{{AHSME box|year=1975|num-b=20|num-a=22}} | {{AHSME box|year=1975|num-b=20|num-a=22}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 20:24, 12 December 2021
Problem
Suppose is defined for all real numbers
for all
and
for all
and
. Which of the following statements are true?
Solution
Let . Our equation becomes
, so
. Therefore
is always true.
Let . Our equation becomes
. Therefore
is always true.
First let . We get
. Now let
, giving us
. Therefore
is always true.
This is false. Let
, for example. It satisfies the conditions but makes
false. Therefore
is not always true.
Since are true, the answer is
.
- mako17
See Also
1975 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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.