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

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1answer
29 views

Integration of characteristic function with varying boundaries

I'm a bit puzzled about integrals with indicator/characteristic functions in them. How do I start computing the following integrals? $$ A\int_{-\infty}^{\infty}f(x)\chi_{[-a+x,a+x]}dx $$ and $$ ...
2
votes
2answers
37 views

Prove that this function is injective

I need to prove that this function is injective: $$f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$ $$f: (x, y) \to (2y-1)(2^{x-1})$$ Sadly, I'm stumbling over the algebra. Here is what I have so ...
0
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2answers
115 views

Rules for combination of odd vs even functional equations

Let $f$ be an even function, and $g$ odd. Let $h$ be some arbitrary function. Is it the case that $f(x) + h(x),\ fh(x),\ hf(x),\text{ and }f(x)h(x)$ are each even or odd according to $h$, and that ...
0
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2answers
62 views

Finding the Asymptote / Root of a reciprocal function

$$y = \frac{3}{8x - 3} $$ The y-intercept is $-1$ and the vertical asymptote is $x = \frac{3}{8}$ but what would be the horizontal asymptote and the x-intercept in this case? I am asking this as the ...
2
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0answers
131 views

Why is the inverse of the Devil's Staircase not measurable?

I recently did an exercise to show that a monotone function $f:X→ℝ $ is Borel measurable (it even only asked for Lebesgue measurability). On the other hand, the inverse of the Devil's Staircase ...
0
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1answer
24 views

Functions (Finding Inverse)

$f(x) = x^2 + 2x$ , domain ${x ≥ 1}$ Question: find the inverse The inverse is $f(x) = 1 + \sqrt(1+x)$ (taking the positive square root only) As $f^{-1}(5) = 2$ and as 2 is an element from the ...
0
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2answers
17 views

Functions inverse + domain

The function $f(x)$ is defined by $f(x)=2x^2-3\quad\{x\in\mathbb R,x<0\}$. Determine (a) $f^{-1}(x)$ clearly stating its domain (b) the values of $a$ for which $f(a)=f^{-1}(a)$. ...
1
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1answer
54 views

If $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$

If a mapping $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$ The mapping $f$ is injective as $f(x) = ...
7
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2answers
186 views

Prove that no function exists such that…

The exercise goes like this: Find a continous function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall c \in \mathbb{R}$ the equation $f(x)=c$ has exactly 3 solutions; Prove that no ...
2
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3answers
63 views

Why is the inverse of this function not a function?

Why does $F^{-1}$ need to be defined on all of $Y$? I can have this function: $g(x)=x,\quad x\ne 3$ and even though it is not defined for all $x$ in its domain, it is still a function, right?
0
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1answer
62 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
1
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1answer
25 views

Something basic; why do I get two different bounds on $f(x) = \frac{x^2}{\sqrt{x^2 + n^{-1}}} + \sqrt{x^2 + n^{-1}}$?

Let $n$ be a natural number. Let $f(x) = \frac{x^2}{\sqrt{x^2 + n^{-1}}} + \sqrt{x^2 + n^{-1}}$. since $x^2 + n^{-1} \geq x^2$, it follows that $$|f(x)| \leq \frac{x^2}{|x|} + \sqrt{x^2 + ...
1
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2answers
35 views

Problem inverting a function

I have this function: $$v(t)=\sqrt{\frac F c} \tanh \left(\frac{\sqrt{Fc}}{m} t \right)$$ I can visually see that t=6.3 when v=27.8, so why don't I get t=6.3 upon putting v=27.8 in this supposedly ...
2
votes
1answer
126 views

Intuition - Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection ...
1
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2answers
42 views

Construct a specific function for a given sequence

Given the sequence $A_n=n$, I want to construct a function $f : R \to R$, such that for every $x \in N$: $f(x)=A_x$ $\int\limits_{0}^{x}f(t)dt=\sum\limits_{i=0}^{x}A_i$ How can do that? Thanks
1
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1answer
63 views

Single variable function derivative w.r.t. time?

I was studying calculus and I had doubts about this problem: (this is not homework) A circular wire expands due to heat so that its radius increases with a speed of $0.01 ms^{-1}$. How rapidly does ...
0
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3answers
34 views

invertibility of $f^{-1}$

In my introductory maths book there is a statement (it follows a theorem) that says:"Note that if f is one-to-one, then $(f^{-1})^{-1} = f$, and so $f^{-1}$ is invertible and also one-to-one because ...
1
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1answer
38 views

How to establish $\sum_{d|n}d\phi(d)$

I am focusing on #5(b). I do not understand how they go from what I have to the answer. Those are r's at the end.
0
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1answer
88 views

Before real numbers are precisely defined, $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$… show $f$ preserves order.

Spivak Calculus, 4th ed., problem 3-17: If $f(x)=0$ for all $x$, then $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x$ and $y$, and also $f(x\cdot y ) =f(x)\cdot f(y)$ for all $x$ and $y$. Now suppose ...
1
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3answers
52 views

Functions definition + question

Am I correct in saying that for Functions, the below is the correct definition: For each value of x in the domain there is only one value of y in the range. Hence, the picture below means that it is ...
0
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1answer
39 views

Showing a function is decreasing

I have $$a_{n} = \left|\int^{(n+1)\pi}_{n\pi} x^{-p}\sin{(x)}~\mathrm{d}x\right|$$ and want to show this is monotonically decreasing, how would I do this? Note $n\in\mathbb{N}$ and $p>0$.
0
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1answer
45 views

How can a function not be one to one and be a function?

My understanding of the definition of a function Given any x, there is only one y that can be paired with x My understanding of a 1 to 1 function Given any y, there is only one x that can be paired ...
0
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2answers
69 views

Range for the function $f(x) = 3x + 2$ with domain $x > 0$

The function below is defined for continuous domains Sketch the graph and state the range of the function Question: $f(x) = 3x + 2$ for the domain $\{x \in \mathbb{R} : x > 0 \}$ The straight ...
1
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2answers
68 views

Problem : Solve $|x^2+x-4| =|x^2-4| +|x|$

Problem : Solve $|x^2+x-4| =|x^2-4| +|x|$ We can find the critical point of each modulus function individually then we get : $x =\pm 2;$ and $x = 0$ $x = \frac{-1 \pm \sqrt{17}}{2}$ So there are ...
2
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1answer
76 views

Roots of Taylor's series.

Show that there is exactly one value of x which satisfies the equation $$2cos^2 (x^3+x)=2^x+2^{-x} $$ I solved this using Taylor's series: $$2^x+2^{-x}=2\{1+\frac {x^2 \{ln2\}^2}{2!}+\frac {x^4 ...
0
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1answer
96 views

Convolution of a continuous function and uniform continuity

Suppose $f$ is a continuous function, and let the convolution $f_n(x) := f \star \varphi_n(x)$ where $\varphi_n$ are smooth test functions. We know $f_n \in C^\infty$. We know that if $f$ is ...
0
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2answers
53 views

For any function $f$ and set $S$, $f(s) \in f(S) \not\implies \Leftarrow s \in S$

This already contains many counterexamples, so I'm not seeking any more of them; I'm interested in learning about my errors with the notation and definitions. Richard Hammack P213 Defintion 12.9: ...
0
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2answers
53 views

Please advise on the order of calculation

I have to make a software making the calculations below over some set of data. That is basically not a problem. The problem I have is with notation of the second formula which is the (Utility ...
-1
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1answer
56 views

Equating coefficients

Excuse me,i don't know how to deal with this problem,i try it for all time of last night, this equation is on "Concrete Mathematics" page 200: d(n) is the number of derangements. e^z is the ...
2
votes
1answer
46 views

What is the use of iterating over a function?

If we have a function, say: $$ f(x) = 3x $$ We can get output values based on linearly increasing input: $$ f(1) = 3(1) = 3 $$ $$ f(2) = 3(2) = 6 $$ $$ f(3) = 3(3) = 9 $$ $$ ... $$ Or, we can ...
2
votes
1answer
136 views

$\textbf{Q}$ fails to prove some correct $\forall$-rudimentary sentence

Show that the existence of a semirecursive set that is not recursive implies that any consistent, axiomatizable extension of Q fails to prove some correct $\forall$-rudimentary sentence. I ...
1
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1answer
29 views

Convert sum to function

I need to convert $\sum_{i=0}^N \frac{C_1}{C_2+C_3i}$, to a function $C_1$, $C_2$ and $C_3$ are constants. I am interested in resulting function itself and method as well.
1
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3answers
370 views

Show that if $f \circ g$ is surjective, then $f$ is surjective, and $g$, the function applied first, needs not to be.

Show that if $f \circ g$ is surjective, then $f$ is surjective, and $g$, the function applied first, needs not to be. (Note:$f \circ g=f(g(s))$, $f$ and $g$ are well defined) This statement originates ...
0
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2answers
25 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
1
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2answers
106 views

Proof strategy for $(=>)$: If $g \circ f = id_A$, then f onto $\iff$ g 1-1. [Chartrand 3Ed P239 9.72]

For nonempty sets A and B and functions f : A → B and g : B → A, suppose that $g \circ f =$ the identity function on A. $(♦)$ (d) $(=>)$ Assume that $f$ is onto. This means there exist ...
4
votes
2answers
85 views

Problem solve functional equation

Solve functional equation:find all strictly monotone functions $f:(0,+\infty)\to(0,+\infty)$ such that $$(x+1)f(\dfrac{y}{f(x)})=f(x+y),\forall x,y>0$$
3
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2answers
135 views

Determine the number of zeros in the first quadrant

This is a homework question: $$f(z) = z^2 - z + 1$$ sorry for the poor code!
2
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2answers
84 views

One to one mapping from $A(S_1)$ into $A(S_2)$

Let $S_1$ and $S_2$ be two sets. Suppose there exists a one to one mapping $\phi$ of $S_1$ into $S_2$. Show that there exists an one to one mapping from $A(S_1)$ into $A(S_2)$, where $A(S)$ means ...
1
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0answers
24 views

How to show the inequation of two function on two different values

Denote $v_1(\lambda)=\frac{(1-\lambda)(2-\lambda)-1}{\sqrt{1+(1-\lambda)^2+((1-\lambda)(2-\lambda)-1)^2}}$ and $v_2(\lambda)=\frac{1-\lambda}{\sqrt{2+(1+\lambda)^2}}$. The figure shows the curve of ...
0
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1answer
29 views

Almost continuity implies measurability?

Trying to prove the continuity of $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ $(n>1)$ I got the following property of $ f $: for all $x\in \mathbb{R}^n $ and $(x_k)$ such that $x_k \rightarrow x$ ...
0
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2answers
77 views

Solve the following equation for real $y$

Ok, here is another problem I also need help with. Solve the equation for real y: $$2\sqrt[3]{(2y-1)} = y^3 +1$$ This is done by defining $$f(y) = \frac{(y^3+1)}{2}$$ So, the equation becomes ...
0
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1answer
52 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
1
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2answers
34 views

Symmetry of product of symmetric functions

Let's say we have three functions, $f(x), g(x) \text{ and }h(x)=f(x)\cdot g(x)$, all of which are defined for $x \in [-1,1]$. If $f(x) \text{ and }g(x)$ are symmetric around the y-axis, will $h(x)$ ...
2
votes
0answers
74 views

Creating a monotonic function

I have $n$ functions $f_i(x) \{i = 1 ,...,n\} $that does not preserve the monotonic mapping order. i.e. if $x_1 < x_2$, then in general, $f_i(x_1)$ is not less than $f_i(x_2)$ (for all $i = 1 ... ...
0
votes
2answers
25 views

Searching for a function

I'm searching for a funcion with this values: f(0)=0 f(1)=1 f(2)=3/2 f(3)=7/4 ... lim from x to infinity: f(x) = 2 I don't want the recursive way to define f.
0
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1answer
53 views

Partial derivatives and functions equal to 0

If I have the function (family of curves) $$F(x,y,p)=(px)^2+p=0$$ I am under the impression that $$\frac{\partial F(x,y,p)}{\partial p}=2px^2+1$$ Is not always equal to $0$. Please could you ...
0
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2answers
35 views

How can i resolve this equation?

Consider the following property $ P(n) $: $ \sum_{k=1}^{n} k = \frac{1}{8}(2n+1)^2 $ Show that $\forall n (P(n) \Longrightarrow P(n+1))$ Where do i start?
1
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4answers
110 views

$f[A]\cap f[B]\supsetneq f[A\cap B]$ - Where does the string of equivalences fail ? [Chartrand 3E 9.12(b), 9.29]

I only realised that equality may fail in $f[A]\cap f[B]\supseteq f[A\cap B]$ (i.e., that we can have $A,B,f$ for which $f[A]\cap f[B]\neq f[A\cap B]$) after checking the answer. I don't see any ...
0
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1answer
29 views

Surjective composed function question

Given two functions $ f $ and $ g $ such that the composite function $ g o f $ is surjective, with respect to the function $ g $ we can state: The function $ g $ is surjective The function $ g $ ...
0
votes
2answers
53 views

Show that $A^{(x,y)}$ is countable.

Question: Let $A$ be a countable set $A^{(x,y)}$ the set of all functions from $(x,y)$ to $A$. Show that $A^{(x,y)}$ is countable. My attempt: By proposition 7.1.2iii, $\mid B \mid^{\mid A \mid}$ ...