# Tagged Questions

For questions about or involving the absolute value function.

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### Proving $\max$ of $a, b$.

How do I prove that $$\max{\{a, b\}} = \frac{a + b + \left | a - b \right |}{2}$$ I have no idea how to even start the proof, any idea / intuition that can get me started is greatly appreciated. ...
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### Square divided by absolute value

First time posting on Math SE, with kind of a basic algebra question. Question Does the relation: $$\dfrac{(ab)^2}{|ab|} = \left|ab\right|$$ with $a,b \in \mathbb{R_{\ne 0}}$ always hold? It seems ...
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### Function with absolute value in denominator - limits

f(x)=(x-1)/(|2-x|-1) |2-x|= { |2-x|; x < +2} {-|2-x|; x >= +2} State domain, range and the equations of the asymptotes. D(f)= {x | x > 3 or x < 3} R(f)= {y | y > 1 or y <= -1} ...
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### Integrating an integrand with an absolute value on exponential

This is one heck of an embarrassment but it is amazing how these bits of subtlety gets lost in the back of the head after the first year of undergraduate studies-with every computation chucked into ...
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### Find all $z$ such that $\left|\tan z\right| = 1$

Find all z such that $$\left|\tan z\right| = 1$$ The first thing that came to my mind was to write tangent in terms of $e^z$ and take its modulus, but I couldn't solve it in this way.
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### Normal distribution question involving absolute value

I don't quite understand how to do the following question. How I tried to do it is to imagine the normal distribution curve, with the highest peak at 4. I understand that |Q| means the absolute value ...
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### Is finding the second derivative of $\sqrt[3]{\vert x\vert}$ the best method to determine if it is convex?

I have an exercise where I have to tell on which intervals a function is concave or convex. I usually do it using second derivative, but I would like to know if there is a simpler way of doing so, ...
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### Is the absolute value of zero positive or negative?

If I had $|x|$, then we know, for pretty much any $x$, that the following is true:$$|x|\ge0$$$$|0|=0?$$Which, by the nature of how we usually apply the absolute value, the solution is positive and ...
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### One absolute value inside of another absolute value in the equation

Let's say we have an equation: $||x|-2| = |2|x|+4|$ How does one go solving it? Symbolab says that it currently doesn't support step by step explanation for this problem, so I would really ...
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### Why is there $(2,3)$ corner coordinates on the graph, using this function: $f(x)=2x-1+|x-2|?$

Currently I am studying about absolute values and I had to consider this function and draw it's graph: $$f(x)=2x-1+|x-2|$$ I helped myself with symbolab, here is the link: ...
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### how to minimize a summation containing an absolute value

Thank you all in advance I have been having some trouble figure out the following problem. You are given a sample {$y_i$}, i=1,… N, from an unknown probability distribution p(y). I want to show the ...
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### Integration of $|e^{-(2+j)t}|^2$

The integration of $|e^{-(2+j)t}|^2$ from zero to infinity is $1/4$ when I separate above as $|e^{-2t}|^2 \cdot |e^{-2jt}|^2$ and integrate. $|e^{-2jt}|$ was taken as $1$. But when I integrate the ...
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### Why does $\sqrt{x^2}$ seem to equal $x$ and not $|x|$ when you multiply the exponents?

I understand that $\sqrt{x^2} = |x|$ because the principal square root is positive. But since $\sqrt x = x^{\frac{1}{2}}$ shouldn't $\sqrt{x^2} = (x^2)^{\frac{1}{2}} = x^{\frac{2}{2}} = x$ because of ...
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### Verify integration of $\int\frac{\sqrt{2-x-x^2}}{x^2}dx$

This is exercise 6.25.40 from Tom Apostol's Calculus I. I would like to ask someone to verify my solution, the result I got differs from the one provided in the book. Evaluate the following integral: ...
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### Find the points at which $f$ has an absolute maximum or minimum on $I$ without graphing

Assume $I=[0.9,3.1], f:I\rightarrow\mathbb{R}$ is defined by $f(x):=|x^2-4x+3|, x\in I$. Without sketching the graph of $f$ on $I$, find points at which $f$ has an absolute minimum on $I$ and points ...
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### What is the name for this operation?

What is the name for this operation? Effectively, take a range and adjust its center point to 0 on a number line. In the case of the example above, I'm specifically looking for the name or ...
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### Some equations involving multiple absolute values

Consider the following equation: $$|x+y^2|+|x-y^2|+|y+x^2|+|y-x^2|=a$$ I'm looking for the method for solving some problems regarding this equation, namely: 1) prove that if $a=2015$, then the ...
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### Silly question about complex numbers - if its modulus is < 1, does raising it to higher exponents make it decrease to the real number 0?

Just playing around with the modulus definition doesn't really confirm that thought... Is it true? If |z| = |x+iy| < 1, is $$\lim_{n\to \infty} z^n = 0 ?$$ Thanks,
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### What exactly is this equation?

Thank you for assistance, I'm just having issues remembering what this is called? For example, the equation would go like this |x+1| = 4 What is this type of equation called, with the two | | ? ...
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### Find extreme values of absolute function

I have to find the extreme values of the following function: $f(x) = |x-2|+|x+3|$ on [-5;5]. How do I do that?
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### Exploring n + abs(x)

In class we are focussing on 'Lattice Theory' at the moment. I have an assignment with instructions saying simply - "Explore the following:" followed by a list of 5 theorems. The purpose of the ...
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### Find the absolute and relative error for a calculator with incorrect rounding

A calculator is out of order. The calculator will round up every single number to the nearest integer if the value at the first decimal digit is 6 and above, or else it rounds down the number to be ...
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### Absolute value of a complex number with a arbitrary basis

I want to calculate the square of the absolute value of a complex number $x^{ia}$, with $x$ and $a$ being real while $i$ is the imaginary number: $$\left|x^{ia}\right|^2=?.$$ I have trouble because ...
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### Function with absolute value and parameters?

I need help with this exercise Consider $f(x)=||x|-a|$ $1)$Determine as $a\in \mathbb{R}$ varies the intervals in which $f(x)$ is continuous and the intervals in which $f(x)$ is differentiable ...
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### Expressing maximum and minimum as $\frac12(x+y)\pm\frac12|x-y|$

I'm looking at ways to get the max/min value of two numbers without using conditional statements, I found these two functions: ...
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### How would you solve an inequality in the form: $|f(x)| < g(x)$?

An inequality such as $|x + 1| < x + 3$ was given to be solved. I attempted to used the theorem: $|x + c| < \delta \implies c-\delta < x < c+\delta$ But $x \in \Bbb{R}$ so $x$ could be ...
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### Initial values problem with absolute value

I've some doubts about initial values problems involving differential equation with absolute values. For example if I have a differential equation like $y'=|x+1|$ with initial condition $y(3)=-2$, ...
The summation of cosine $\sum_{k=1}^N \cos (k x)$ is well known (for example, see the previous question here) and is called Lagrange's trigonometric identity. Is it possible to construct a similar ...
### Prove that $|ab+1|>|a+b|$ with $|a|<1$, $|b|<1$
Prove that $|ab+1|>|a+b|$ with $|a|<1$, $|b|<1$ $a$, $b$ are real numbers Where $|a|$ is the absolute value of $a$. Every time, I arrive to a dead-end.