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

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0
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1answer
11 views

Invalid range from inequality

We were given this function and asked to give Range. $$f(x)~=~\dfrac{x^2}{x^2+1}$$ Now I took 3 cases and deduced that $\text{Range} = \left[~0,\infty ~\right)$ Now it is obvious that if we divide ...
0
votes
1answer
23 views

Inverse of elementary functions

which may be two right inverse of: 1) $h:\Re \rightarrow [0,\infty) $ defined by $h(x)=|x|$ 2) $k:\Re \rightarrow [1,\infty)$ defined by $k(x)= e^{x^2}$
4
votes
2answers
32 views

Domain of the given function

A function $y(x)$ is defined as $$ 2^y+2^x=2 $$ The question is about finding it's domain. Pretty simple. By observing the function I could say all the negative numbers are in the domain. But, I think ...
0
votes
1answer
15 views

Cardinality of Sets and injections

Let A,B,C,D sets. if |A| $\le$|B| and |B| < |C|, show that |A| < |C| Proof: Case1: suppose |A| < |B| then there exists injection f: A$\to$B and |B| < |C| then there exists injection ...
1
vote
2answers
35 views

Taking a time derivative of a function of 3 variables.

I have a function of $3$ variables which are all functions of $t$. $$x = \frac{v_1t-y}{\sqrt{(v_2/\dot{x})^2 -1}} \tag 1 $$ In the equation $v_1,v_2$ are constant and $x$ and $y$ are both function ...
2
votes
2answers
53 views

Confusion with seeming lack of notational coherence between $\sin^{-1}(x)$ and $\sin^2(x)$

It seems that $\sin^2(x)$ is used to denote the square of whatever value $\sin(x)$ is, instead of the expected $(\sin(x))^2$. Based on that, I would assume that $\sin^{-1}(x) = \frac{1}{\sin(x)}$, ...
1
vote
4answers
61 views

limit of $\ln x + (x+1)/x$ as $x$ approaches $o$

I want to establish monotone intervals of function $f:(0, \infty) \rightarrow \mathbb R$, where $f(x)=(x+1)\ln x$ using its first derivative. I proved that the first derivative of $f$ is an injective ...
-3
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1answer
23 views

Assume that p is a real number . In order for $\sqrt[3]{x+3p+1}-\sqrt[3]{x}=1$ to have real solutions, then p [on hold]

Assume that $p$ is a real number. In order for $\sqrt[3]{x+3p+1}-\sqrt[3]{x}=1$ to have real solutions, then $p$: Options A) $p \geq 1/4$ B) $p \geq -1/4$ C) $p\geq1/3$ D) $p \geq-1/3$
0
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4answers
58 views

Suppose $f$ is a real function satisfying $f(x+f(x))$ = $4f(x)$ and $f(1) = 4$. Then the value of $f(21)$?

Should I proceed with just putting the value of $f(1)=4$ in the first equation or there will be a different way of solving this ?
0
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1answer
17 views

Let $\{a,b,c,d,e,f,g,h\}$ be distinct elements in the set $\{ -7 , -5 , -3 , -2 , 2 , 4 , 6 , 13 \}$ .

Let $\{a,b,c,d,e,f,g,h\}$ be distinct elements in the set$ \{ -7 , -5 , -3 , -2 , 2 , 4 , 6 , 13 \}$ . The minimum possible value of $(a+b+c+d)^2 + (e+f+g+h)^2$ is ? I tried it by making the first ...
1
vote
0answers
29 views

A special limit-sum interversion

Let $f$ be a function and $(F_k)_{k\in\mathbb{N}}$ a sequence of functions converging simply to $f$. Let $a_{n,k\in\mathbb{N}}$ be such that $\displaystyle\forall ...
1
vote
1answer
51 views

Can a function be differentiable while having a discontinuous derivative?

Recently I came across functions like $x^2\sin(1/x)$ and $x^3\sin(1/x)$ where the derivatives were discontinuous. Can there exist a function whose derivative is not conitnuous, and yet the function is ...
1
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2answers
33 views

$\sqrt{4x -3}$ injective? Bijective? Inverse?

I'm shown part of the function $g(x) = \sqrt{4x-3}$. Is it injective? I said yes as per definition if $f(x) = f(y)$, then $x =y$. Is this right? Under what criteria is $g(x)$ bijective? For what ...
1
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2answers
17 views

confusion about solving and graphing a simple rational function

given the function: $\frac{x+1}{5} - 2 = -\frac{4}{x}$ I could multiply through by $5x$ yielding the quadratic with solutions $(5,4)$: $x^2 - 9x + 20 = 0$ or.... I could create a common ...
0
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1answer
13 views

Proof that $f:\mathbb CG \to S$ given by $f(a)=a\cdot s$ is surjective

I am going through a proof of the following result from my lecture notes: Suppose $\mathbb C G \cong \bigoplus_{i=1}^nU_i$ where the $U_i$ are simple $\mathbb CG $ modules. Then any simple $\mathbb ...
0
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2answers
17 views

Function (How to determine limit by using Maclaurin's series?)

$$ f(x)=\begin{cases} \dfrac{\sin x^2}{1-\cos2x},&-\pi<x<0 \\[1ex] \dfrac{2+\sqrt{x}}{4-\sqrt{x}},&0\leq x<\pi & \end{cases}$$ My question is how to determine whether ...
2
votes
3answers
38 views

Construct a non-linear function that shows that the intervals $[2,4]$ and $[10,22]$ have the same cardinality

Using something other than a linear function, show the intervals $[2,4]$ and $[10,22]$ have the same cardinality. I don't quite know where to start with this problem, or what key factor is necessary ...
1
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3answers
119 views

Finding a convex function between two points

Given two points of the $xy$ plane, is there a way to find the equation of a convex function between those two points? I know the answer wont be unique so I'm just looking for a general equation that ...
0
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2answers
41 views

Show that the function $f:\mathbb{Z} \rightarrow \mathbb{2Z}$ by $f(a)=a^3+3a+2$ is not onto

If the function $f:\mathbb{Z} \rightarrow \mathbb{2Z}$ by $f(a)=a^3+3a+2$, then show that $f$ is not onto. Hint: Show that $f(a)\neq 0$. I have a feeling I have to use the root theorem test, but I ...
1
vote
4answers
39 views

Domain of the function $f(x) = \sqrt{\frac{3^x-4^x}{x^2-4x-4}}$ will be?

I tried solving this question by $1.$ $-1$ and $4$ will not be in domain because denominator can not be zero . $2.$ Either both denominator and numerator will be positive or negative so that whole ...
2
votes
3answers
23 views

Investigating the bijectivity of $ 2 x + |\cos(x)| $.

The question asks if the function $$ f(x) = 2 x + |\cos(x)| $$ if (one-one, onto), (many-one, onto) or (one-one, into). After a long process of plotting the graph, I managed to guess it’s one-one and ...
2
votes
4answers
53 views

A formal proof that the function $ x \mapsto x^{2} $ is continuous at $ x = 4 $.

Problem: Show $f(x)=x^2 $ is continuous at $ x = 4$. That is to say, find delta such that: $ ∀ε>0$ $ ∃δ>0 $ such that $ |x-a|<δ ⇒ |f(x)-f(a)|<ε$ Where $a=4$, $f(x)=x^2$,and $f(a)=16$. ...
0
votes
2answers
14 views

Proof of Injection and Surjection

I am having trouble proving the function f is injective and surjective. $f$ is a function from $\mathbb{Z}\times{Z} \to $\mathbb{Z}\times{Z}$ and $f(x,y) = (5x-y,x+y)$. I know it should be fairly ...
1
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0answers
21 views

How we get the result of this limit?

I met a problem while doing my homework. Let say we have a formula: $(a_3 + d)\cdot sin(\theta_2) - b_3 \cdot cos(\theta_2) - a_2 = 0$ Now we knew $a_3$, d(in this case is exactly 0), $b_3$ and ...
0
votes
1answer
15 views

Functions problem: surjectivity and direct and inverse image theory

I need some help with this problem, if sombody could give me any idea of how to solve it (not the solution itself, but it would be better) I will appreciate it: for a function $f: A → B$, prove $ ∀ Z ...
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votes
1answer
39 views

Prove that $f:\mathbb N\to\mathbb Z$ is a bijection. [on hold]

Define $f:\mathbb N\to\mathbb Z$ by $$ f(n) = \begin{cases} (1-n)/2& \text{if $n$ is odd;}\\ n/2 &\text{if $n$ is even.}\end{cases}$$ Prove that $f$ is a bijection and determine the ...
-3
votes
1answer
30 views

Prove that α : p(z) -> p(z) is a bijection [on hold]

Please help me prove this: Let $\alpha : \mathscr{P}(\mathbb{Z}) \rightarrow \mathscr{P}(\mathbb{Z})$ be defined by $$\alpha(S)=\begin{cases} S\cup\{0\} \text{ if }0\notin S\\ S\setminus\{0\}\text{ ...
1
vote
0answers
25 views

How do I find the domain and range of this piecewise defined function?

Are both conditions true when $x>3$? If so, how do I graph it? $$ f(x)=\left\{\begin{aligned} &x^2-4&&:x>3\\ &2x-1&&:x\geq 3 \end{aligned} \right. $$
0
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4answers
64 views

if $f:X \to Y$ is 1-1 and $|X| = |Y|$, does that imply $f$ is onto?

Similarly, if $f$ is onto and both sets have the same cardinality, does that imply $f$ is 1-1? I'm pretty sure both statements are true but I'd rather not assume. Thank you for your time.
1
vote
2answers
30 views

Correct to write $\vec{F}:\mathbb{R}^3\rightarrow\mathbb{R}^3$?

Suppose I have some vector field \begin{align} \vec{F}\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)&=G\textbf{i}+H\textbf{j}+T\textbf{k}.\tag{1} \end{align} Would it be correct for ...
1
vote
1answer
40 views

Nice Formula for a Function from $\mathbb{N}\cup\{0\}$

I am trying to get a nice formula for the following function $$f:\mathbb{N}\cup\{0\}\rightarrow \{1,1,-1,3,-3,5,-5,7,-7,9,-9,...\}$$ thus It seems like it would be closely related to somthing like ...
0
votes
0answers
10 views

How should I create a single score with two values as input?

I have two series of values, a and b as inputs and I want to create a score, c, which reflects both of them equally. The distribution of a and b are below In both cases, the x-axis is just an ...
0
votes
1answer
8 views

Is the piece-wise function with mapping $f: (\mathbb{Q^{+}_r} \cup \{0\}) \rightarrow \mathbb{Z^+}$ injective or surjective?

Let $f: (\mathbb{Q^{+}_r} \cup \{0\}) \rightarrow \mathbb{Z^+}$ by $\begin{array}{cc}\Bigg \{&\begin{array}{cc} f(0)=1 \\ f(\frac{a}{b}) = a+b \end{array} \end{array}$ where $\mathbb{Q^{+}_r} = ...
2
votes
2answers
46 views

How to smoothly approximate a sign function

I have a function that defined as following $$f(x) = \begin{cases} 1, & \text{if $x > 0$ } \\ 0, & \text{if $x=0$ } \\ -1, & \text{if $x<0$ } \end{cases}$$ In practice, the $f(x)$ ...
0
votes
1answer
15 views

Prove that the relation $f$: ($\mathbb{Z^*} \times \mathbb{Z^*}$) $\rightarrow$ $\mathbb{Q}$ by $f(a,b)$ = $\frac{a+b}{a+b-3ab}$ is a function

Prove that the relation $f$: ($\mathbb{Z^*} \times \mathbb{Z^*}$) $\rightarrow$ $\mathbb{Q}$ by $f(a,b)$ = $\frac{a+b}{a+b-3ab}$ is a function I believe that f is a function and I am attempting to ...
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votes
2answers
33 views

How do I evaluate these functions [on hold]

(1) $(f\circ g)(x)=3x^2$ and $g(x)=3x^2-1$. What is $f(x)$ (2) $(f\circ g)(x)=9x^2+6x$ and $g(x)=3x+1$, find $f(x)$.
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votes
0answers
10 views

Formula for a recursive function

Given the recursive function $T: \mathbb{N}_0 \to \mathbb{R}_+$ $$T(n) ≤ max_{1 ≤ k ≤ n - 1}\{T(k-1) + T(n - k) + c n^2\}$$ with c > 0 constant, I want to determine an absolute formula to quickly ...
13
votes
8answers
1k views

Can we have a one-one function from [0,1] to the set of irrational numbers?

Since both of them are uncountable sets, we should be able to construct such a map. Am I correct? If so, then what is the map?
0
votes
2answers
37 views

What is the range of the function $f(x)=\log x+\sin x $?

I've been thinking about this problem for some time now and I initially thought that the range was $\mathbb R$ . I arrived at that conclusion because in a similar problem we had to find the range of ...
1
vote
1answer
12 views

When can you not do a mapping composition?

Suppose I have $\alpha:\mathbb R^3 \to \mathbb R$ and $\beta:\mathbb R \to \mathbb R^+$. Looking over my notes, it says $\alpha \circ \beta$ can not be done but $\beta \circ \alpha$ can. What is the ...
3
votes
2answers
54 views

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$ What I did : Use $g(x) = f(x)-x$, then $g'(x) = f'(x)-1\ne0$ I suspect I have to use Rolle's theorem now, But I am having difficulty ...
9
votes
3answers
297 views

Finding a pair of functions with properties

I need to find a pair of functions $f$, $g$ such that $f$ is not differentiable at $x = 0$ $g$ is not differentiable at $f(0)$ $g \circ f$ is differentiable at $x = 0$ I've tried a lot of ...
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votes
0answers
30 views

math question that I can't figure out. [on hold]

Mikhail saved \$1 of his paycheck the first week. He saved \$2 the second week. He saved \$4 the third week. He saved \$8 the fourth week. If this pattern continues, how much will he save the fifth ...
0
votes
1answer
30 views

Is $f:\mathbb{Q^*} \rightarrow \mathbb{Q}$ by $f(\frac{a}{b}) = \frac{\max{(a,b)}}{\min{(a,b)}}$ a function?

Suppose that the relation $f:\mathbb{Q^*} \rightarrow \mathbb{Q}$ by $$ f \Bigl(\frac{a}{b} \Bigr) = \frac{\max{(a,b)}}{\min{(a,b)}} $$ is defined. Then is $f$ a function? If so, how would we prove ...
2
votes
1answer
15 views

The domain of a function as a function: the “domain-function”

The domain of a function $f:X\to Y$ is normally defined as $\operatorname{dom}f\equiv X$, but I would like the domain-function $\operatorname{dom}$ to be a funtion itself, i.e. I would like to define ...
0
votes
0answers
15 views

Question Concerning a General Method to Show that a Relation is Not a Function

When showing that a relation is not a function, is there an efficient method for finding particular preimages that are mapped to more than one image, rather than attempting to find particular ...
2
votes
1answer
32 views

Simple true/false statements about function composition

Given the functions $f,g,h$ from $\mathbb{R}$ to $\mathbb{R}$ I have to determine whether the following statements are true: "If $f \circ g$ is strictly increasing and $f$ is injective then $g$ is ...
1
vote
3answers
55 views

What the terms “basis functions” and “orthogonal” denote in the case of signals?

I am a beginner. I have read that any given signal whether it is a simple or complex one, can be represented as summation of orthogonal basis functions. Also, there are many ways through which basis ...
2
votes
0answers
46 views

Is there a way to follow the curve of a polynomial at a fixed speed?

I'm trying to follow the curve of a polynomial between $x=0$ and $x=1$ at a fixed speed using the arc length formula: $\int_0^1\sqrt{1+f'(x)^2}dx$ I've gotten around the square root problem by ...
0
votes
1answer
13 views

Is the relation $\Psi: M_2(\mathbb{Z^*}) \rightarrow \mathbb{Q}$ a Function

Determine if the following is a function Let $\Psi: M_2(\mathbb{Z^*}) \rightarrow \mathbb{Q}$ by $\Psi\big( \left[\begin{smallmatrix} a&0\\ 0&b\end{smallmatrix}\right]\big) = \frac{a}{b}$ ...