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

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2
votes
0answers
75 views

increasing and decreasing functions with functional equations

If $f'(\sin x)<0$ and $f''(\sin x)>0$ for all $x$ between $(0,\frac{\pi}{2})$ and $g(x)=f(\sin x)+f(\cos x)$, then find the set of values of $x$ for which $g(x)$ is decreasing between $(0,\frac{\...
1
vote
3answers
75 views

Why is $f(n)=n^2+3$, where $f\colon\mathbb{N}\to\mathbb{Z}$, not an onto function?

Question: $f_2 :\mathbb{N} \to \mathbb{Z}, f_2(n)=n^2 +3$ Using algebra, making $y=f(n)$, isolating for $n$ and plugging in the expression back, I get $n$. However, the answer key says it is not onto....
0
votes
2answers
35 views

quick function proof

If $f(x)$ is a function with unit area, show that the scaled and stretched function $\frac{1}{a}f(\frac{x}{a})$ also has unit area. Before you give an answer I would still like to try to prove it ...
0
votes
2answers
36 views

Finding the point at which a function is continuous

I am trying to understand the solution to the following question. At which $c\in\mathbb{R}$ is the function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined by $$f(x)=\begin{cases}x&\text{if $x$ is ...
1
vote
4answers
63 views

Functions: If $f(g(x))$ is onto, does this mean $g(x)$ is onto

Question: Let $g:A \to B$ and $f:B \to C$ be two functions. If $f$ and $f \circ g$ are onto, is $g$ necessarily onto? I know it's not, but I don't understand why/don't know how to explain it.
1
vote
2answers
47 views

Vertical asymptotes of a given non-rational radical funtion

We have that $f$ is a function $f(x) = x\sqrt{x+4}$. Hence, $f'(x) = \dfrac{3x+8}{2\sqrt{x+4}}$. Then, $\lim_{x \to -4^+}f'(x) = -\infty$. This means that $f$ has a vertical slope at $f(-4)$. It ...
0
votes
1answer
36 views

For the 2nd derivative of $f(x)$ to be continuous, does $f(x)$ need to be continuous?

For example: $f(x) = \begin{cases}\arctan(x) & |x| < 1 \\ \frac{x^3}{12}+\frac{3x}{4} & |x|\ge1 \end{cases}$ where $f''(x) = \begin{cases}-\frac{2x}{(x^2+1)^2} & |x| < 1 \\ -\frac{...
0
votes
3answers
37 views

Let $f:\mathbb N\to\mathbb N$, be $f(n)=\left\lfloor\frac{2n+2}3\right\rfloor$

Let $f:\mathbb N\to\mathbb N$, be $f(n)=\left\lfloor\frac{2n+2}3\right\rfloor$. Is f one-to-one? Is f onto? I found it quite easy to find a counter-example for $f$ not being one-to-one. $f(2)=f(3)$, ...
0
votes
1answer
604 views

How can I tell if a function is a monotonic transformation?

In my Economics class, we are talking about monotonic transformations of ordered sets. But I don't understand how I can tell if a given function will preserve the order. My Question What ...
1
vote
0answers
19 views

Proper name for the inverse of a mutivariate function w.r.t. one variable

Given the function $f(x,y,z): \mathbb{R}^3 \rightarrow \mathbb{R}^1$, what is the proper name of the function $g(w,y,z): \mathbb{R}^3 \rightarrow \mathbb{R}^1$ such that $f\big(g(w,y,z),y,z\big)=w$?
0
votes
1answer
45 views

Prove that $f_{n}$ is not uniformly convergent

Let $\{f_{n}\}$ be this sequence of functions: $f_{n}(x)=nx$ when $0\leq x\leq \frac{1}{n}$, $f_{n}(x)=2-nx$ when $\frac{1}{n}<x<\frac{2}{n}$ and 0, when $\frac{2}{n} \leq x \leq 1$. I have to ...
-1
votes
1answer
96 views

Rewrite the absolute value of a rational function as a piecewise defined function

How do I rewrite the function : $f(x)=\left|\frac{2x^5 - x^2 + 1}{x^2 - 4}\right|$ as a piecewise defined function? The absolute value around both the numerator and the denominator is what's ...
7
votes
2answers
356 views

A differentiation question conceptual query

I'm quite unsure about how to deal with differentiation of absolute functions, and their continuity. For example, the question I was dealing with was the following: $$ f(x) = \frac{x}{1 + |x|}$$ ...
0
votes
3answers
44 views

Working out recursive functions

I know that a function is defined recursively when it calls itself to progressively converge on a solution based on an initial condition. For example if we consider the function defined by $$Sq(1) = ...
0
votes
1answer
29 views

Find $\sup_{x\in(0,\alpha)} \frac{n x^2}{x^3+n^3}$.

Find $$\sup_{x\in(0,\alpha)} \frac{n x^2}{x^3+n^3}$$ where $\alpha>0$ and $n\in\mathbb N$. I found $\sup$ doing the derivative. Is there an alternative way (without derivative)? Thank you very ...
0
votes
1answer
44 views

Write an inductive proof that if there is a surjection $ f : \lceil m \rceil → \lceil n \rceil $ then $m ≥ n$.

Here's the problem: Write an inductive proof that if there is a surjection $ f : \lceil m \rceil → \lceil n \rceil $ then $m ≥ n$. Where I Am: I assume that I should induct on $n$ and come to the ...
0
votes
2answers
41 views

Let $m,n \in \mathbb N^+$. Define an explicit bijection from the Cartesian product $\lceil m \rceil \times \lceil n \rceil$ to $\lceil mn \rceil$.

Here's the problem: Let $m,n \in \mathbb N^+$. Define an explicit bijection from the Cartesian product $\lceil m \rceil \times \lceil n \rceil$ to $\lceil mn \rceil$. My Progress: Obviously, I'm ...
0
votes
1answer
49 views

Prove that there is a surjection $ f:X \to Y$ if and only if $ |Y| \le |X| $.

Here's the problem: Let $X$ and $Y$ be sets. Prove that there is a surjection $$ f:X \to Y$$ if and only if $$ |Y| \le |X| $$. My work so far: I am working on the following direction: If $ |Y| ...
0
votes
1answer
151 views

How to transform function values to specific interval

I'm doing a project at university about scientific computing and I'm stuck. As in: I seem to lack quite a bit of mathematical background for this project. The program has as input an array of $x$ ...
1
vote
2answers
36 views

Constant Moving Speed

I've made this graph: https://www.desmos.com/calculator/czk3ylyokj As you can see, the purple point is slowing down as it approaces the extreme point. How can I make this point move with constant ...
-4
votes
1answer
31 views

Find values of a and b if the function is continuous

First I took left hand limit and right hand limit but I am not sure how to simplify the equations and remove the 0/0 form . Looking for your help desperately. Have an exam tomorrow help is appreciated ...
4
votes
2answers
456 views

English for “prolongement” oder “Fortsetzung”?

I'm sorry if that's not the right place to ask for, wikipedia failed to give me the correct word... What's the english for a function that is defined on a larger domain than the original function and ...
4
votes
3answers
122 views

$F(x)+G(y)= e^{x+y}?$

Are there functions $F(x)$, $G(y)$, such that $F(x)+G(y)=e^{x+y}$ , where $x,y$ are real numbers? I have been trying all elementary functions, and have no clues on what else I could do.
3
votes
3answers
316 views

Mean Value Theorem Inequality Contradiction(?)

I am trying to show: $e^x > 1+x+\frac{x^2}{2}$ for $x>0$ using Mean Value Theorem (MVT). My method is as follow: consider the function $f(x)=e^x - (1+x)$. We know $f'(x) = e^x -1 > 0 $...
0
votes
1answer
22 views

Unique solution of nolinear equation set

$$\left\{ \begin{aligned} f_1(x_1,x_2...x_n)=0 \\ f_2(x_1,x_2...x_n)=0 \\ \vdots \\ f_n(x_1,x_2...x_n)=0 \end{aligned} \right. $$ $f_i\in C^\infty(R^n)$,what is the condition that make the equation ...
1
vote
1answer
250 views

Show that the metric space C[a,b] is complete. [duplicate]

Prove that the metric space $C[a,b]$ is complete. Where $C[a,b]$ is the collection of continuous $f:[a,b] → R$ and $||f|| = sup_{x \in [a,b]} |f(x)|$, such that $\rho (f,g) = ||f - g||$ is a metric ...
1
vote
3answers
185 views

How can I prove the inverse of a function is odd?

Given function $f$, where $A⊆ \mathbb{R} $ is a symmetric domain with respect to 0,$ \;\; f:A \rightarrow\mathbb{R}$ and $f$ is an odd one-to-one function, I need to prove $f^{-1}$ is odd. I was ...
1
vote
1answer
64 views

Suppose that $f : \mathbb{R}\rightarrow \mathbb{R}$ is continuous and that $f(x)\in \mathbb{Q}$ for all $x \in \mathbb{R}$. Prove $f$ is constant. [duplicate]

I am really stuck on this proving this statement, so could someone please help me get through this. Thank You. P.S. This can be proved through elementary analysis results instead of going into ...
-1
votes
4answers
305 views

How do I prove that $\arccos(x) + \arccos(-x)=\pi$ when $x \in [-1,1]$? [closed]

Prove that $\arccos x + \arccos(-x) = \pi$ when $x \in [-1,1]$. How do I prove this? Where should I begin and what should I consider?
1
vote
4answers
132 views

Range of $f(x) =\frac {x -1}{x^2 -2x + 3} $?

Is my solution for finding the range of $$f(x) = \frac{x-1}{x^2 -2x + 3} $$ correct? Since its Domain is $ \mathbb{R} $, so transforming this equation into $x$ in terms of $y$ , we get $$ yx^2 - (2y +...
0
votes
1answer
28 views

bounds on binomial coefficients

Do the standard upper bounds on the binomial coefficient $\binom{n}{k}$ still work well if $k=f(n)$ (by standard i mean for example $(\frac{en}{k})^{k}$ and $\frac{n^{k}}{k!}$)? In particular if $k=o(...
1
vote
0answers
40 views

Verifying proof that $f$ has an inflection point at zero if $f$ is a function that satisfies a given set of hypotheses.

Prove that if $f$ is a function $f(x): f'(x) > 0$ $\forall x: x \neq 0 \wedge f'(0) = 0 \wedge f''$ is a continuous one to one function on some open interval $(a, b): a < 0 < b$ then $f$ ...
1
vote
1answer
198 views

Find if relation is reflexive, symmetric or transitive

Let $A = \{1, 2, 3, 4\}$ and let $F$ be the set of all functions from $A$ to $A$. Let $R$ be the relation on $F$ defined by "For all $f, g$ in $F$, $(f, g)$ in $R$ if and only if $f (i) = g (i)$ for ...
1
vote
3answers
352 views

How to evaluate this exponential fraction limit?

I am trying to determine if 3$^n$ grows faster than 2$^{2n}$. One way I found online to do this was, from Growth was to evaluate $\lim_{n\to \infty} \frac{3^n}{2^{2n}}$ and if that limit evaluates ...
13
votes
8answers
1k views

Why are vector valued functions 'well-defined' when multivalued functions aren't?

I'm looking for an 'intuitive' answer here, because I have no formal mathematical training but find myself in a comparatively math-heavy PhD (visual perception; lots of neuroscientists on the one side ...
0
votes
1answer
80 views

How to denote the minimal bounding curve of two intersecting function curves?

If there are two functions $y=f(x)$ and $y=g(x)$, $x,y\in\mathbb{R}$, how could I denote the minimal bounding curve of these functions? (See the green dotted line on the figure.) I'm looking for an ...
0
votes
1answer
73 views

Non-constant Linear Boolean Function

How can we prove that any non-constant linear Boolean function is balanced ? I know that any non-constant affine function is balanced. But i cannot expend this for Boolean function.
0
votes
1answer
36 views

Simplifying trigonometric equations

Here is the question: simplify the expression $$\frac{\sin(f+g)+\sin(f-g)}{\cos(f+g)+\cos(f-g)}.$$ For this questions, are all of the addition and subtraction identities of sin and cos required? I ...
0
votes
1answer
44 views

Definition of Lebesgue integrable function

If a function $f : \mathbb{R}^d \to [-\infty,\infty]$ is Lebesgue integrable, then by definition we have $$\int_{\mathbb{R}^d} |f(x)| \, dx < +\infty.$$ Is it possible to say that there exists a ...
0
votes
2answers
114 views

Solving Trigonometric Identities - Thinking questions

The question I have is a thinking question: If $\sin(x+y)=0.9$ and $\sin(x-y)=0.6$, determine $\sin x \cos y$. I am really not sure how to go about it. Could I use the addition formula of sin and the ...
0
votes
1answer
52 views

Series of random numbers on a continuous function

At one point, I read about a function used to generate random numbers that follow a continuous pattern. By this I mean random numbers that as a series is random, but in which items tend to be ...
0
votes
0answers
39 views

Let $J⊆I⊆\Bbb R$ be open intervals, let $c∈J$, and let $f:I\setminus\{c\}→\Bbb R$ be a function.

Let J⊆I⊆$\Bbb R$ be open intervals, let $c∈J$, and let $f:I\setminus\{c\}→\Bbb R$ be a function. Prove that $\lim_{x \to c}⁡f(x)$ exists if and only if $\lim_{x→c}⁡f/_{J}(x)$ exists, and if these ...
4
votes
2answers
224 views

Function, Relation, Operation and Cartesian Product

An operation is a kind of function. A function is a kind of relation. A relation is a subset of a Cartesian product. A Cartesian product is an operation. Back to 1. It seems to me that there's ...
4
votes
4answers
77 views

$f :\mathbb N \to \mathbb R$ be the function $f(0)=0 , f(n)=\dfrac 1 n , \forall n >0$;is $\mathbb N$ induced with the metric $|f(x)-f(y)|$ compact?

Let $\mathbb N$ be the set of non-negative integers and $f :\mathbb N \to \mathbb R$ be the function $f(0)=0 , f(n)=\dfrac 1 n , \forall n >0$ , then obviously $f$ is injective , so $d : \mathbb N \...
2
votes
1answer
137 views

Constructing a quasiconvex function

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$ f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, \...
1
vote
4answers
75 views

Proving trigonometric identity $1+\cot x\tan y=\frac{\sin(x+y)}{\sin x\cos y}$

$$1+\cot x\tan y=\frac{\sin(x+y)}{\sin x\cos y}$$ I have worked through most of this question, and I believe I am so close to finding the answer, but I have run into some issues where I am not sure ...
1
vote
2answers
70 views

limits involving a piecewise function, Prove that if $c \ne 2$, then f does not have a limit at $x = c$.

$$f(x) = \begin{cases} (x-2)^3 & \text{if $x$ is rational } \\ (2-x) & \text{if $x$ is irrational } \end{cases}$$ (i) Prove that if $c \ne 2$, then f does not have a limit at $x = c$. (ii) ...
0
votes
3answers
37 views

Prove the trigonometric Identity involving secant

The question I am currently working on is: $\sec^2x-2\sec x\ \cos x+\cos^2x=\tan^2x-\sin^2x$. Okay, judging by the expression here, I am going to need to work with the left side of the equation first,...
0
votes
2answers
38 views

$\sin x \sin(\pi/2-x)=\sin x \cos x $?

During a question involving proving a trigonometric identity, I was given help in which one of the lines showed that $\sin x \sin(\pi/2-x)$ equals $\sin x \cos x$? Could anyone please explain to me/...
1
vote
2answers
63 views

Proving Trig. Identities [duplicate]

$\cot x=\sin x \sin(\pi/2 -x) + \cos^2x \cot x$ I'm having difficulty with figuring out how to prove trigonometric identities. I know that in order to do these you need to use the trig ratios ...