1
vote
3answers
144 views

Is there a function whose derivative is $|x|$?

Is there a function $y=f(x)$ such that $$\frac{df}{dx}|_{x=a} =|a|$$ for all $a\in \mathbb R$? I'm in a debate with my friend over it and we are stuck
2
votes
2answers
51 views

Finding the limit of $F(x)=\frac{x^2-4}{|x+2|}$

Let $F(x)=\dfrac{x^2-4}{|x+2|}$ and find the following limits $(a) \; \; \lim_{x \to -2^-}F(x)=$ $(b) \; \; \lim_{x \to -2^+}F(x)=-4$ $(c) \; \; \lim_{x \to -2}F(x)=DNE$ I substituted $-2$ to find ...
4
votes
2answers
51 views

Piecewise linear function and absolute value

While writing a solution to homeworks for my students, I had to write the function $$f(x)=\left\{\begin{array}{ll} \frac{x+2}{2}, & x\leqslant -4\\ \frac{x}{4}, & -4\leqslant x\leqslant 4 \\ ...
2
votes
2answers
29 views

Concerning Rules of Exponents & Absolute Value

I understand that one of the accepted definitions of the absolute value function is $\left| x \right| = \sqrt{x^2}$. However, I do not understand why if I substitute $-5$ in for $x$ that I can't do ...
0
votes
1answer
22 views

Modulus function (working out coordinates)

Lets say you have $y = -|3x - 1|$ when working out where it cuts the axis, particularly the x-coordinate you do the following when $y = 0, 3x - 1 = 0$ therefore $x = 1/3 $ the modulus and the ...
0
votes
3answers
26 views

What's the best method to graph the following function by hand…

Here in my exercise I have to study the function and draw its graph. Can you please tell me what's the best method to do this, because I don't think that's reasonable to use the input output method, ...
1
vote
2answers
146 views

Suppose f(x) is an odd function. Prove that g(x) = |f(x)| is an even function.

Suppose f(x) is an odd function. Prove that g(x) = |f(x)| is an even function. I understand that an odd function is where f(-x) = -f(x), and an even function is where f(-x) = f(x), but am struggling ...
0
votes
1answer
33 views

Absolute value being an odd function

Correct me if I am wrong, but I learned that for a function to be symmetrical to the origin, it can be rotated 180 degrees and still appear the same. How is ${x^2 - y^2 = 0}$ an odd function if when ...
0
votes
5answers
94 views

Why is $f(x)=|x|$ not differentiable?

Consider the function $f(x)=|x|$, I know that $f$ is not differentiable at $x=0$, but still, when you try to differentiate $f(x)=\sqrt{x^2}$ (which is exactly the same), you get: ...
0
votes
1answer
85 views

Steps to Graph Exponential Equations & Absolute Value

how to sketch: $-e^{|-x-1|} + 2$ Can someone clarify: $|f(x)|:$ we draw $f(x)$ and then reflect the ($-y$ parts) in the $x$-axis $f|(x)|:$ we draw $f(x)$ and then reflect the ($-x$ parts) in the ...
-1
votes
4answers
59 views

A function where absolute maximum is also absolute minimum?

What is an example of a real-valued function where an absolute maximum is also an absolute minimum?
4
votes
4answers
109 views

Let $x$ be in the set of real numbers $\mathbb{R}$ and let $f(x)=|2x-1|-3|2x+4|+7$ be a function, write $f(x)$ without the absolute value.

Let $x$ be in the set of real numbers $\mathbb{R}$ and let $f(x)=|2x-1|-3|2x+4|+7$ be a function, write $f(x)$ without the absolute value. I thought of it this way: $$f(x)=\begin{cases}2x-1-3(2x+4)+7 ...
-2
votes
1answer
130 views

Example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points [duplicate]

So I had an exam today and one of the questions were: Give an example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points. At first I had no idea how to do it ...
1
vote
1answer
48 views

When is $|f(x)|$ equivalent to $f(|x|)$

Specifically for functions of a complex variable. Are there any rules of thumb?
1
vote
1answer
260 views

Finding domain of a rational function

Find the domain and graph: $$f(t)=\frac{-t}{|t|}$$ My book says to define it piecewise. My questions: $\mathbf{1)}$ Do all rational functions have to be defined piecewise, or just this ...
1
vote
2answers
928 views

Proof the maximum function $\max(x,y) = \frac {x +y +|x-y|} {2}$ [duplicate]

I want to prove the maximum function max: $\mathbb{R} \rightarrow \mathbb{R}$, which is defined by $$\max(x,y) = \begin{cases}x, \text { if } x \geq y , \\ y, \text { if } x < y \end{cases}$$ ...
1
vote
2answers
567 views

How to find minimum of sum of mod functions?

How to find minimum value of $$|x-1| + |x-2| + |x-31| + |x-24| + |x-5| + |x-6| + |x-17| + |x-8| + \\|x-9| + |x-10| + |x-11| + |x-12|$$ and also where it occurs ? I know the procedure for find answer ...
13
votes
6answers
1k views

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
1
vote
2answers
552 views

Express $y=|-x^2+1|$ as a piecewise function.

I'm unsure of how to start this problem. Any help would be greatly appreciated.
0
votes
2answers
67 views

Prove That $|a +b| = |a| +|b|$ if $a$ and $b$ Have Same Signs, And $|a +b| < |a| + |b|$ if $a$ and $b$ Have Opposite Signs (Proved Differently) [duplicate]

My Proof: This problem has mainly four cases, they are as follows: 1) $a, b > 0$ 2) $a, b < 0$ 3) $a > 0 > b$ 4) $a < 0 < b $ Let suppose that the sum of the real numbers $a ...
0
votes
5answers
112 views

Prove That $|a +b| = |a| +|b|$ if $a$ and $b$ Have Same Signs, And $|a +b| < |a| + |b|$ if $a$ and $b$ Have Opposite Signs

My Proof: $|a +b| = |a| +|b|$ ..... $(i)$ $|a +b| < |a| + |b|$ ..... $(i)$ If $'a'$ and $'b'$ have same signs: Let $a$ and $b$ be equal to $-x$. Replacing $a$ and $b$ with $-x$ in the equation ...
2
votes
2answers
5k views

How to find critical points of an absolute values function

I am asked to find How many critical points does the function $g(x) = |x^2 − 4|$ have? I know that the result is $3$ but I can only find $2$. What I do, is to equal the equation to $0$, so $x^2-4=0$ ...
-1
votes
2answers
166 views

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?
0
votes
0answers
65 views

Name for $\max(x, \frac 1x)$, $x > 0$? [duplicate]

Possible Duplicate: Multiplicative Identity analog for absolute value In looking for the most extreme scaling, I'm comparing $f(a)$ and $f(b)$ where $f(x) = \max(x, \frac 1x)$, $x > 0$? ...
1
vote
1answer
7k views

Limit with absolute value

I found this limit within the Calculus Single Variable book from Thomas. $$ \lim _{x \to -2^-} (x+3) \frac{|x+2|}{(x+2)}$$ This is how I'm trying: First of all, we need to found where the absolute ...
1
vote
4answers
211 views

Understanding $y=|mx+n|$

The diagram shows the graph of $y=|mx+n|$ (i tried my best to do the same thing as my exercise book, actually 1 is propotional to 1 and 3 is propotional to 3, but 2 is not propotional to 2) Find ...