Elementary questions about functions, notation, properties, and operations such as function composition.

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2
votes
1answer
56 views

Determining position with respect to time

If we assume $s(t)$ as a time-dependent position function and $v(t)$ as a time-dependent velocity function($v = \frac {\mathrm{d}s} {\mathrm{d}t}$) and $v=8\sqrt{s}$, how could I determine $s$ with ...
1
vote
5answers
134 views

Can the roots of $f(x)=x^4-x^3+2x^2-x-1$ be found algebraically?

Can the roots of $f(x)=x^4-x^3+2x^2-x-1$ be found algebraically? Are there multiple methods for doing so?
1
vote
0answers
38 views

Optimize multi-step calculation into one step?

Prerequisites The following helper functions are defined: Condition $$ \delta (x,y) = 0^{(x-y)^{2}} $$ This is a simple condition, it returns 1 when x = y. It will be used to modify products to ...
0
votes
3answers
68 views

Solving the functional equation $f(a + b) = a + f(b)$

How would you solve: $$f(a+b)=a+f(b) ?$$ It seems similar to the Cauchy equation $$f(a+b)=f(a)+f(b),$$ but I'm not sure what to do with this. I have a feeling the only solution is $f(k)=k$ but idk. ...
0
votes
0answers
43 views

Trying to prove that if $f:[a, b]\to[s, t]$ is monotone then $f$ is continuous

I'm trying to prove that if $f:[a, b]\to[s, t]$ is monotone (and its image is closed interval) then $f$ is continuous. My attempt: I say wlog, $f$ is increasing. I know that a monotone function only ...
0
votes
0answers
45 views

Local minima or maxima of conditional function

How do I find the local minimum/maximum of this function? $f:{R-2}\to{R}$ $f(x)=\frac{1}{x-2}e^{\left|x\right|}$ I wrote it like this $$f(x)=\left\{\begin{array}{cc} \frac{1}{x-2}e^x & x \ge 0 ...
1
vote
2answers
520 views

Can any function represent something in the real world?

We know that the volume of a cube can be represented by the function: $V(x)=x^3$, where $x$ is side length. $x^2$ can represent the volume of some material that has a constant side ($1$). The function ...
1
vote
1answer
74 views

Finding the minimum value of a function.

Find the minimum value of the function: $$f(x) = \frac{\left(x+\frac{1}{x}\right)^6-\left(x^3+\frac{1}{x^3}\right)^2 - 2}{\left(x + \frac{1}{x}\right)^3 +\left(x^3 + \frac{1}{x^3}\right)}$$ for $x&...
2
votes
1answer
61 views

Graph of a non-continuous function is closed

Exercise. Let $\ f\colon\mathbf R\to\mathbf R$ be defined by $$f(x)=\begin{cases}\frac{1}{x},\ x>0\\0,\ x\leq 0\end{cases}$$ Prove that the graph $\Gamma_f:=\{(a,f(a)):a\in\mathbf R\}$ of $f$ ...
0
votes
2answers
52 views

Unbiased estimator.

With sample variance defined as $S^2 = (\sum_{i=1}^n (X_i - \overline{X})^2 )/(n-1)$ A. Show that $E(X_i^2) = \sigma^2 + \mu^2$ using the fact that $\sigma^2 = E((X_i - \mu)^2)$
1
vote
6answers
176 views

Minimizing a function - sum of squares

I'm hoping you can help with this problem. I haven't taken calculus in years and I don't know where to start... The sum of squares of a sample of data is minimized when the sample mean is used as the ...
2
votes
1answer
116 views

Find all function $f: \mathbb{Q} \to \mathbb{Q}$ such that $f(x+f(x)+2y)=2x+2f(f(y))$ [closed]

Find all function $f: \mathbb{Q} \to \mathbb{Q}$ such that $f(x+f(x)+2y)=2x+2f(f(y))$ for all $x,y \in \mathbb{Q}$.
0
votes
1answer
54 views

Differentiability issue with this function

$f:D\to{R}$ $$f(x)=\frac{1}{x-2}e^{\left|x\right|}$$ Find the domain $D$ of the function and study whether the function is differentiable. Find the left and right derivatives in the points where the ...
0
votes
1answer
50 views

Can anyone help out with this proof, certain steps are unclear. Norm of linear operator.

I have the following norm defined as follows (in $R^n$, $x=(x^1,x^2,\ldots,x^n)\ $)$\| x\|_1= \sum_{i=1}^{n}|x^i|$ Let $A:R^m \to R^n$ a linear map of the spaces $(R^m ,\| \cdot \|_1 )$ and $((R^n ,\|...
-4
votes
1answer
66 views

Why is this statement true? [closed]

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be the function $f(x,y)=(y-x^2)(y-2x^2).$ Why is this statement true: $t\mapsto f(t\xi)$ has in $t=0$ a local minimum for every $\xi\in\mathbb{R}^n$
0
votes
1answer
31 views

Each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$ when $f$ is additive

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an additive function such that $f(1)=0$. Then for each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$. Since $f$ is additive and ...
0
votes
1answer
26 views

Determine relation of $x$ and $y$ from results

I can't seem to determine the relation between $x$ and $y$ for this problem. All of the previous ones I have done have been doable simply by eye-balling the relation between $x$ and $y$, but here I am ...
1
vote
2answers
176 views

What is the meaning of the notation of function

More specifically, what is meant by the function $T: \mathbf V \to \mathbf W$? I saw it in the discussion of linear map in Axler's Linear Algebra Done Right but could not understand this notation. ...
2
votes
4answers
132 views

What is the difference between these two statements involving minimums of a function?

All values at which $f$ has a local minimum. and All local minimum values of $f$.
3
votes
2answers
54 views

Expressing a function in terms of compositions of three functions.

Express the function F in the form $f \circ g \circ h$. $$F(x)=\frac {9}{( x^2 + 7)}$$ I'm not sure how to get $x^2+7$ in the denominator. Here is what I tried: $$h(x) = (x+7)$$ $$g(x) = x$$ $$...
0
votes
3answers
41 views

Possible values of 'a' ? $f(x)=(x^2+ 2 ax +a^2-1)^{\frac{1}{4}}$

If $$f(x)=(x^2+ 2 ax +a^2-1)^{\frac{1}{4}}$$ has its domain and range such that their union is set of real numbers,then what should be the possible values of a? What can be the approach?
1
vote
2answers
63 views

Finding the inverse of the function $f(k, x) = k^{x}x.$

Recently, I have been looking at the function $f(x) = e^{x}x,$ where its inverse is the Lambert W function. I was intrigued by the fact that it is rather hard to calculate its solution, in comparison ...
0
votes
0answers
31 views

Calculating the local minimum of a function

Regarding this function $f(x,y)=1007x^2-x^{2014}+(e^y-1+2x^2)^2$. I want to find the strict local minimum of $f$. I started calculating $\nabla$: $\nabla (1007x^2-x^{2014}+(e^y-1+2x^2)^2)$ $=(2014x+...
0
votes
1answer
32 views

Determining continuity and differentiability

Is this function continuous and differentiable? $$f(x)=\left\{\begin{array}{cc} 1-x & x<1 \\ x^2-2x+1 & x\:\ge 1 \end{array}\right.$$ For continuity, I did $$\lim_{x\to 1^+\:}f(x) = \lim_{...
9
votes
1answer
86 views

If a set $S$ has a choice function, does $\bigcup S$ have one too?

I have an exercise in a book that asserts that if a set $S$ has a choice function on it, then so does the union of all its elements $\bigcup S$ (without assuming the axiom of choice). I, however, have ...
0
votes
1answer
53 views

What does it mean for a “formula to be undefined”?

I was covering the techniques used sketch rational functions of five different types as follows: However, then I encountered this: And, Ij just can't find out what it means for the formula to ...
1
vote
2answers
64 views

Analysis of continuity and differentiability of a function

Find a,b,c $\in \mathbb{R}$ for which the function is a) continuous, b) differentiable. $$f(x)=\left\{\begin{array}{cc} ax^2+bx+c & x<0 \\ 2\sin x+cos x & x\:\ge 0 \end{array}\right.$$ ...
5
votes
4answers
964 views

What kind of growing function has a constant as limit?

My knowledge in mathematics are a bit old and I'm looking for functions with constant as limit. The function must always grow. The curve should be something similar to $\sqrt{x}$ or $\ln(x)$ but with $...
4
votes
5answers
357 views

Showing a function is injective using that $f'(x)\ne0$

Given a differentiable function $f\colon \mathbb R\to\mathbb R$ which we must prove to be injective, does it suffice to show $f'(x)≠0$ for all $x$ (for which the function is defined)? It makes sense, ...
6
votes
4answers
115 views

Showing a function $f: \mathbb{N} \times\mathbb{N} \to \mathbb{N}$ is injective

Let $f: \mathbb{N} \times\mathbb{N} \to \mathbb{N}$ with $$ f(i,j) = \frac{(i+j-2)(i+j-1)}{2}+j. $$ I want to show $f$ is an injection. This is how I approached the problem: I tried to show $f(i,j)...
4
votes
1answer
63 views

Does the concept of “cograph of a function” have natural generalisations / extensions?

First, definitions: The graph of a function $f : A \to B$ is a subset of $A \times B$, namely the set $\{(x,y) : x \in A, y \in B, f(x) = y\}$. The cograph of a function $f : A \to B$ is the ...
0
votes
1answer
45 views

Find the range of this function [closed]

How do you find the range of the following function, please? $$\frac{2x^2 + 20}{x^2 + 5}$$
0
votes
0answers
52 views

Complexity of computing a posiform of a quadratic pseudo-boolean function

I am reading the chapter 13, Pseudo-Boolean functions, of Boolean Functions: Theory, Algorithms, and Applications by Crama et. al. In section 13.2, the authors introduce the idea of Posiform. The ...
1
vote
3answers
40 views

Evaluating a function at a point where $x =$ matrix.

Given $A=\left( \begin{array} {lcr} 1 & -1\\ 2 & 3 \end{array} \right)$ and $f(x)=x^2-3x+3$ calculate $f(A)$. I tried to consider the constant $3$ as $3$ times the identity matrix ($3I$) but ...
0
votes
0answers
46 views

Is it possible to define an inverse of the main three trig. functions without domain restrictions?

Ok, I know that the main three main trigonometric functions, that is the tangent, sine, and cosine, are periodic and thus not one-to-one, but onto. And, since an inverse requires a function to be onto ...
0
votes
1answer
52 views

Explain why this composite function is not allowed?

Explain why this composite function is not allowed when $f(x) = 2x+1, x \in [-5,5]$ and $g(x) = x^2, x \in \mathbb{R}, x \geq 0$ How would you change the domains so that the function $fg(x)$ can ...
5
votes
3answers
154 views

Can we obtain $f(y+x)=y+f(x)$ from $f(x^2+f(x)^2+x)=f(x)^2+x^2+f(x)$?

Find all function $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that $$f(m^2+f(n))=f(m)^2+n.$$ Let $P(x,y)$ be the assertion: $f(x^2+f(y))=f(x)^2+y \; \forall x,y \in \mathbb{Z}^+.$ $P(x,x)$ gives ...
1
vote
1answer
77 views

Confused about basic of image

Hello I tried to work a problem from the text called " Introduction to Real Analysis" by Robert G Bartle and Donald Sherbert and I encountered a small difficulty. I am starting to think that my ...
3
votes
4answers
61 views

What does $f^{-1}(B)= \{ x \in X \mid f(x) \in B\}$ mean?

I have encountered the expression $$f^{-1}(B) = \{ x \in X \mid f(x) \in B\}$$ My questions are: 1) What does the $-1$ exponent mean in this context? 2) Is it right to say "if the set $X$ ...
4
votes
1answer
48 views

Prove that $f(X\cap f^{-1}(Y))=f(X)\cap Y$

Let $\ f\colon A\to B$ and let $X\subset A$, $Y\subset B$, prove that $$f(X\cap f^{-1}(Y))=f(X)\cap Y$$ The "$\subset$"$-$inclusion is easy: if $y\in f(X\cap f^{-1}(Y))$, exists a $x\in X\cap f^{-...
3
votes
2answers
87 views

What is the actual definition of a function?

I am learning precalculus and my book defines the following: A function $f$ from a set $A$ to a set $B$ is a rule that assigns to every element $a$ in $A$ one and only one value in $B$. Well, I ...
0
votes
1answer
62 views

What is the function $f$ such that $\sum_{k=0}^n f(k)=n^3$?

$$\begin{align*} 1 &\leadsto 1 \\ 1+3 &\leadsto 2^2 \\ 1+3+5 &\leadsto 3^2 \end{align*}$$ In general, if $f(x)=2x+1$, then $f(0)+f(1)+f(2)...f(n)=(n+1)^2$. Now, $$\begin{align*} 1 &...
1
vote
1answer
17 views

Find formula structure for a complex function

I am looking to find the function formula structure of a repeating function like the one in the image linked below.... Something that repeats indefinitely (like a sine wave) on the X-axis. Anybody ...
-1
votes
1answer
66 views

Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $h=f-g$ [closed]

Please please please please please I want some help ,Is there and body here who can help me in this question : Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $...
3
votes
1answer
322 views

If the derivative is zero on [a, b], then the function is constant - using Heine-Borel?

I know the proof of the statement in the title using the mean value theorem but I was wondering if it can be proved using the Heine-Borel Lemma, which reads: "Every open cover of a closed interval ...
-1
votes
2answers
73 views

Let $f(x)$ be a polynomial such that $f(a)=b, f(b)=c, f(c)=a$ Then Prove that $a=b=c$. [closed]

Let $f(x)$ be a polynomial in $x$ With integer coefficient. If for natural numbers $a,b,c$, $f(a)=b, f(b)=c, f(c)=a$ Prove that $a=b=c$.
0
votes
1answer
52 views

well defined mapping-function

I would like to know how to show an mapping or function is well defined i think in generale we use that : -$f$ is well defined mapping iff $( x\in E\implies f(x)\in F)$ in particular when mapping ...
1
vote
2answers
70 views

Is 'a' differentiable in f when f is a product of a differentiable and non-differentiable function?

Recently, I was studying differentiable and non-differentiable functions and I wondered whether this "conjecture" of mine is true: 1) "If $f(x)$ is a function that is the product of $g(x)$ and $h(x)$ ...
1
vote
3answers
57 views

$f(f(y)+1)=y+f(1)$ is bijective.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(xf(y)+x)=xy+f(x), \; \forall x,y \in \mathbb{R}.$$ I read a solution in finding this function. It states that setting $x=1$ gives ...
1
vote
2answers
33 views

Can we say that a function is increasing/decreasing on some range if there's a vertical asymptote in that range?

The graph below shows the function $f(x)=\frac{e^x}{x-1}$ Can we say that the function is decreasing for all $x\le2$ (there's a local minimum at $x=2$) or do we have to take the asymptote at $x=1$ ...