# Questions tagged [absolute-value]

For questions about or involving the absolute value function.

2,023 questions
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### Log laws and modulus

If you have the log of a modulus, (like after integration), how do the log laws work? So if you have $a\ln\left|2x-3\right|$ does it become: $\ln\left|(2x-3)^a\right|$ or $\ln(\left|2x-3\right|)^a$, ...
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### Absolute function continuous implies function piecewise continuous?

I have a simple true/false question that I am not sure on how to prove it. If $|f(x)|$ is continuous in $]a,b[$ then $f(x)$ is piecewise continuous in $]a,b[$ Anyone that can point me in the ...
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### True/false question: limit of absolute function

I have this true/false question that I think is true because I can not really find a counterexample but I find it hard to really prove it. I tried with the regular epsilon/delta definition of a limit ...
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### double integral of an absolute function

I'm just a little unsure of how to tackle this one. I understand that typically you would separate the integral into two for where x is positive or negative, I'm just unsure of how to separate it for ...
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### The relationship between the derivative of $f(x)$ and $|f(x)|$

I have seen it in an exercise book. I don't know how to do it. If $f(x)$ is continuous at $x=a,$ and $|f(x)|$ can be differentiated at $x=a,$ then $f(x)$ is differentiable at $x=a.$
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### I can't find a absolute value function that have [-1,1] range

I want a function $f:\mathbb{R}\to[-1,1]$ with absolute value like $f(x)=|a-x|\ldots$ that have $[-1,1]$ range. Can anybody help me?
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### The solution set of the equation $|2x - 3| = - (2x - 3)$

The solution set of the equation $\left | 2x-3 \right | = -(2x-3)$ is $A)$ {$0$ , $\frac{3}{2}$} $B)$ The empty set $C)$ (-$\infty$ , $\frac{3}{2}$] $D)$ [$\frac{3}{2}$, $\infty$ ) $E)$ All real ...
$e^{\frac{2}{x-1}\log\left|x-1\right|}+1\neq 0$ Since that this is an exponential function, this equation is verified $\forall x \in \mathbb{R}$? Or I have to consider the absolute value of the ...
$$\frac{\sqrt{4 + \arccos\left|\frac{2-x}{x+3}\right|}}{\sqrt{x^2 - 4x + 5} - 3}$$ I'm trying to find the natural domain of the function above. I set up this conditions:  \begin{cases}\sqrt{x^2 ...