For questions about or involving the absolute value function.

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4
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3answers
69 views

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. ...
0
votes
3answers
42 views

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 ...
0
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0answers
25 views

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} ...
0
votes
0answers
36 views

Show that absolute value satisfies triangle inequality, how? [duplicate]

I wish to show that given $a,b,c \in \mathbb{R}$, the following holds: $|a - c| \leq |a-b| + |b-c|$ Using the definition $| x | = \max\{x, -x\}$ I can't seem to be able to show that $|a - c| \leq ...
2
votes
6answers
97 views

Solving the absolute value inequality $\big| \frac{x}{x + 4} \big| < 4$

I was given this question and asked to find $x$: $$\left| \frac{x}{x+4} \right|<4$$ I broke this into three pieces: $$ \left| \frac{x}{x+4} \right| = \left\{ \begin{array}{ll} ...
2
votes
2answers
62 views

Evaluating the Fourier coefficients of $abs(x)$

Let's get started: $$\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi} |x|e^{-inx} dx$$ since $|x|$ is an even function: $$= \frac{1}{\pi}\int_0^{\pi} xe^{-inx} dx$$ Integration by parts yields: ...
0
votes
0answers
23 views

Continuity properties of an example function $f:\mathbb{R}^n\to\mathbb{R}$

Consider the function $f:\mathbb{R}^n\to\mathbb{R}$ defined as follows: $$ f(x)=\begin{cases} ||x||^2 & \text{if $||x||\le 1$,}\\ 1/||x||^2 & \text{if $||x||> 1$,} \end{cases} $$ where ...
0
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3answers
43 views

Nested absolute-value inequality

I try to solve a problem in two ways, but the results are not the same. Method 1. $$\lvert \lvert x \rvert + x \rvert \le 2$$ For $x < 0$, we have $\lvert x \rvert = -x$. Therefore: $$\lvert ...
0
votes
1answer
28 views

Modulus of Two Complex Numbers, Squared

I have a very silly question to ask! I have $|z_{1} + z_{2}|^2 = |z_{1}|^2+|z_{2}|^2+2|z_{1}||z_{2}|\cos{\theta}$, where $z_{1}$ and $z_{2}$ are complex numbers. For the life of me I cannot ...
1
vote
1answer
56 views

Prove those inequalities are true [duplicate]

I want to prove that those inequalities are true for $a, b ∈ R$: $$ |a + b| ≤ |a| + |b| $$ $$ ||a| − |b|| ≤ |a − b| $$ $$ |a − b| ≤ |a − c| + |c − b| $$ Now I can see that they are true, and I could ...
0
votes
1answer
34 views

Proving inequality with absolute value [duplicate]

How can I show the following inequality for any real numbers $x,y,z$? $$\frac{|x-z|}{1+|x-z|}\le \frac{|x-y|}{1+|x-y|} + \frac{|y-z|}{1+|y-z|}.$$ The triangle inequality could be useful, but I am ...
0
votes
1answer
22 views

Which of the two following solutions is correct for absolute value of this expression?

I'm currently going through Spivak and ran across this problem, but i see a difference in my answer and the answer that i'm checking it again. The problem is to eliminate the absolute value signs in ...
5
votes
6answers
160 views

Prove that $|-x| = |x|$

Using only the definition of Absolute Value: $\left|x\right| = \begin{cases} x & x> 0 \\ -x & x < 0 \\ 0 & x = 0,\end{cases}$ Prove that $|-x| = |x|.$ This seems so simple, but I ...
3
votes
2answers
115 views

Derivative of $x\cdot|x|$ on $x=0$?

$$f(x) = x |x|$$ Wolfram Alpha says is: $$f'(x) = \frac{2x^2}{|x|}$$ and thus $f'(0)$ is indeterminate, while an HP48 says that: $$f'(x) = |x| + x \operatorname{sgn} x,$$ which would yield $f'(0) ...
1
vote
1answer
34 views

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 ...
2
votes
3answers
106 views

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.
0
votes
1answer
28 views

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 ...
3
votes
2answers
59 views

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, ...
0
votes
5answers
90 views

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 ...
0
votes
2answers
30 views

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 ...
2
votes
2answers
36 views

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: ...
1
vote
1answer
43 views

How Does This Line Intersection Equation Work?

For the lines: $y = ax + b$ $y = cx + d$ The standard intersection equation $x_i = \frac{d - b}{a - c}$ $y_i = \frac{ad - bc}{a - c}$ If the points that I was given to find the line $y = ax + ...
1
vote
3answers
80 views

Integrating $f(x)=\int|\cos(x)|dx$ and then solving $f(x)=\frac {2x}{\pi}$?

I realised the other day that by applying absolute value signs to the cosine function and then integrating, I would get an almost sine function that doesn't have negative slope. And then I also ...
0
votes
1answer
24 views

Validity of inequalities using integrals and absolute value

This question is similar to this one but the only response was pointing out mistakes in the solution. My goal is to determine whether the operator $T: C[0,1] \to C[0,1]$ defined by $Tx = ...
0
votes
1answer
36 views

Integrating absolute terms

This is just to clarify my doubt regarding absolute values functions. Lets say there is a function $$f(x) = ax^{2} - \left|\frac{bx}{c}\right|$$ and we are asked to integrate this over $-\infty \to ...
3
votes
2answers
42 views

Absolute value equation with rational expression

I am to solve the equation: $|\frac{2x}{x^2 - 3} | < 1$ And so: 1. I rewrote it as $|2x| < |(x - \sqrt 3) | |(x + \sqrt 3)|$ And I tried to divide it into a few intervals For $ ...
1
vote
1answer
65 views

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 ...
1
vote
2answers
35 views

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 ...
5
votes
4answers
146 views

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 ...
5
votes
1answer
166 views

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: ...
1
vote
2answers
24 views

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 ...
1
vote
4answers
193 views

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 ...
1
vote
1answer
16 views

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 ...
1
vote
1answer
33 views

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,
0
votes
4answers
47 views

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 | | ? ...
0
votes
1answer
16 views

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?
7
votes
2answers
620 views

A ''strange'' integral from WolframAlpha

I want integrate: $$ \int \frac{1}{\sqrt{|x|}} \, dx $$ so I divide for two cases $$ x>0 \Rightarrow \int \frac{1}{\sqrt{x}} \, dx= 2\sqrt{x}+c $$ $$ x<0 \Rightarrow \int \frac{1}{\sqrt{-x}} \, ...
0
votes
1answer
52 views

Double absolute value inside integral

Any ideas as to go about doing this particular integral? $$\int\limits\limits_{-1}^{4}||x^2+x-6|-6| dx$$ I'm a bit confused as to how to consider the cases into account. My idea was to consider 4 ...
1
vote
2answers
55 views

If $\lim_{x \to x_0} f(x) = L$, then $\lim_{x \to x_0} \lvert f(x)\rvert = \lvert L \rvert$.

If $\lim_{x \to x0} f(x) = L$, then $\lim_{x \to x0} \lvert f(x)\rvert = \lvert L \rvert$. I know this is true, because $\lvert f(x) \rvert - \lvert L \rvert <= \lvert f(x) - L \rvert < ...
0
votes
1answer
34 views

complex absolute value equations

How on earth does one solve this problem? I know that to solve abs value equations we have to consider both the negative and postive possibilities, but the constant (6) makes this a little more ...
3
votes
2answers
53 views

Property of function $\varphi(x)=|x|$ on $\mathbb{R}$

Define $\varphi(x)=|x|$ on $[-1,1]$ and extend the definition of $\varphi(x)$ to all real $x$ by requiring that $\varphi(x+2)=\varphi(x).$ How do you prove that for any $s,t$ $$ ...
0
votes
0answers
29 views

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 ...
-1
votes
1answer
73 views

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 ...
1
vote
3answers
27 views

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 ...
0
votes
1answer
24 views

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 ...
0
votes
1answer
35 views

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: ...
0
votes
2answers
22 views

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 ...
2
votes
2answers
43 views

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 $y(3)=-2$, ...
2
votes
0answers
130 views

Summation of the absolute value of the variable

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 ...
2
votes
2answers
64 views

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.