Absolutely Wrong

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An absolute value on $\mathbb Q$ is a function $f(x)$ that satisfies the four following axioms:

i. $f(x) > 0$ for all $x$

ii. $f(x)=0$ if and only if $x=0$

iii. $f(xy) = f(x)f(y)$

iv. $f(x+y) \geq f(x)+f(y)$

$f(x)=\left| x \right|$ obviously satisfies these conditions. Which of the following functions also satisfy these four axioms?

$f(x) = x$
$f(x) = x^2$
$f(x) = e^x$
$f(x) = 100 \left| x \right|$
$f(x) = \sqrt{x}$
$f(x) = \begin{cases} 0 & \text{if x=0} \\ 1 & \text{if x=1} \end{cases}$

The answer will be the number of the function that follows the four axioms. If more than one of the functions satisfy the four axioms, type their sum. For example, if you think the answer is number 2, 4, and 5, the answer will be $2+4+5=11$ .

The answer is 6.

1 solution

Joshua Lowrance
Oct 22, 2019

	$f(x)>0$	$f(x)=0$ if and only if $x=0$	$f(xy)=f(x)f(y)$	$f(x+y) \geq f(x) + f(y)$
$f(x)=x$	No	Yes	Yes	Yes
$f(x)=x^2$	Yes	Yes	Yes	No
$f(x)=e^x$	Yes	No	No	No
$f(x)=100 \left\| x \right\|$	Yes	Yes	Yes	No
$f(x)=\sqrt{x}$	Yes	Yes	Yes	No
$f(x)=\begin{cases} 0 & \text{if x=0} \\ 1 & \text{if x=1} \end{cases}$	Yes	Yes	Yes	Yes

Only the sixth formula fits as an absolute value.

Absolutely Wrong

The answer is 6.

1 solution

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