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

learn more… | top users | synonyms (1)

0
votes
2answers
20 views

How to find the values of a at which $y$ is increasing?

I don't know how to solve this one and the question is: Find the values of a at which $y = x^3 + ax^2 + 3x + 1$. My solution is: $y'= 3x^2 + 2ax + 3$ I know that if $y' \ge 0$, $y$ should be ...
2
votes
0answers
18 views

Order the domain so that function is monotonic

Let $f : \mathbb{R} \to \mathbb{R}$ be a measurable function. Is there a bijection $b: \mathbb{R} \to \mathbb{R}$ such that $f \circ b$ is monotonic?
-1
votes
1answer
26 views

What is the function between x and y? [on hold]

X --- Y 4 | 1 3 | 2 2 | 3 1 | 4 0 | 5 I hope I am wording it right, basically I want to get my y using my x.. sooo Y = X _____
1
vote
1answer
26 views

If f is continuous and strictly increasing, then the function $f^{-1}:f(I)\rightarrow I$ is continuous and strictly increasing.

Let $I \subseteq \Bbb R$ be a non-degenerate open interval, and let $f:I\rightarrow \Bbb R$ be a function. Suppose that f is strictly monotone. If f is continuous and strictly increasing (or ...
0
votes
2answers
53 views

Is it possible to find the value of $x$ where $e^x$ exceeds $x^{10}$ by hand?

All I managed is to "simplify" the equation $e^x=x^{10}$ to $\frac{x}{\ln{x}}=10$. Is there some way or trick to make the equation look like $x=\dots$? (Solve the equation, in other words.)
-3
votes
0answers
21 views

How to determine whether injective or surjective over this functions [duplicate]

G : N×N given by G = 2x+5 ∀x ϵ N H : Z×Z given by H = 10 ∀x ϵ Z I got an idea whether injective or surjective but don't know how to go through. And finally, are these functions? I think they are ...
-1
votes
1answer
19 views

How to go towards this functions and defining whether injective or surjective

G(x) : N×N given by G(x) = 2x+5 ∀x ϵ N H(x) : Z×Z given by H(x) = 10 ∀x ϵ Z I am not familiar with this notations. However, I got an idea whether injective or surjective. And finally, are ...
6
votes
5answers
42 views

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing.

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing. I do not have use derivatives, so I decided to apply the definition of being a strictly ...
3
votes
1answer
42 views

Are there only a few 'universally convergent' Taylor Series?

The taylor series for $sin(x)$, centered at any point, converges for all x. The taylor series for $e^{x}$ and $cos(x)$ do as well. Thus, taking an algebraic function of these (without division) ...
0
votes
3answers
59 views

Beginner level : What is the intuitive meaning and what are the steps to prove for an injective function

For the proof that a map is into, it is convenient to use the contrapositive of the definition of one-to-one, namely: $$ \forall x,y \in X, f(x) = f(y) \rightarrow x = y. $$ where the definition of ...
6
votes
2answers
120 views

What is the meaning of $\mathbb{R}\setminus\{0\}$?

This is used in many posts related to functions and googling it doesn't help. What does this mean? $\mathbb{R}$ should stand for all Real numbers.
3
votes
5answers
60 views

Number of functions for $(f(x))^2=x^2$

If $f\colon\mathbb{R}\to\mathbb{R}$ is a function such that $$(f(x))^2=x^2$$ for all $x$ , then 1) The number of such functions are? 2) How many of them are continuous? I can see 4 functions: ...
0
votes
0answers
24 views

proof that some expected value equal to $\theta (\log n - \log k)$

So here is the problem - Given the following equation: $(c_2\cdot \log n) - (c_1\cdot \log k)\le E(X)\le 1+ (c_1\cdot \log n) - (c_2\cdot \log k)$ When $c_2,c_1\gt0$ and also $c_1\gt c_2$ In ...
1
vote
4answers
23 views

Is the mapping $f\colon\mathbb{R}\to\mathbb{R}$ defined by $f(x)=5x^3+3$ onto?

Let $f\colon \mathbb R \to \mathbb R$ be defined by $f(x)= 5x^3+3$. Is it onto? According to me, if $y=5x^3+3$, then $x = \sqrt[3]{(y-3)/5}$ is not an element of $\mathbb R$ for all $y \in ...
1
vote
1answer
16 views

functions and the commutative property

with regard to vector spaces of functions. How do I know if the commutative property holds for a set of functions. especially if the vector space includes an infinite set. for instance, for the ...
1
vote
2answers
43 views

Why the continuity of $f$ is not a necessary condition?

I am quite new to functions and continuity, and now I am reading the slides regarding the intermediate value theorem, which is related to continuity of functions. While reading, I found the ...
3
votes
2answers
18 views

Increasing/Decreasing intervals of a parabola

I am being told to find the intervals on which the function is increasing or decreasing. It is a normal positive parabola with the vertex at $(3,0).$ The equation could be $y = (x-3)^2,$ but my ...
1
vote
1answer
40 views

Find $f(x)$ given $f, g$ such that $\,f(0) =2,\, g(0) =1, \, f'(x) = g(x),\, g'(x) = f(x)$.

Let $f$ and $g$ be functions satisfying: $$\begin{align} f(0) & =2\\ g(0) &=1 \\ f'(x) &= g(x) \\ g'(x) & = f(x) \end{align}$$ Find $f(x)$.
-1
votes
0answers
13 views

Marginal distribution for two continous random variables [on hold]

Let $f_{XY} (x,y) = a\;if\: x ∈ [-1, 0] \;and\; y ∈ [x^2, -x] $ or $f_{XY} (x,y) = 0 \;if\;x \notin [-1, 0] \;and\; y \notin [x^2, -x] $ is the distribution function for the random variables X and Y . ...
0
votes
1answer
19 views

What is $h^{-1}(L)$, for $L$ a regular language and $h$ a homomorphism?

Let $L = L((00 + 1)∗)$ and $h : \{a, b\}^* \to \{0, 1\}^*$ be defined by $h(a) = 01$ and $h(b) = 10$. What is $h^{−1}(L)$? In this context "$+$" means "$\cup$". So the language $L$ is all the ...
3
votes
1answer
61 views

Functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$

How to find all functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$ for $x> 1$?
2
votes
1answer
20 views

given the following two conditions, find $f(x,y)$

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions: $f(x+t,y)=f(x,y)+ty$; $f(x,t+y)=f(x,y)+tx$; $f(0,0)=k$; then for all $x,y \in\mathbb R$, $f(x,y)=$ a) ...
1
vote
1answer
31 views

domain of $\sqrt {\cos^{-1}(\cos x)-\lfloor x\rfloor} $

Here is my question where I got stucked. The domain of $\sqrt{\cos^{-1}(\cos x)-\lfloor x\rfloor} $ where $\lfloor \cdot\rfloor$ denotes the greatest integer function (floor function).
0
votes
0answers
9 views

Lagrange Multiplier, Boundary

In many cases when we have to optimize a function under a constraint, i.e $f(x,y)=e^{-xy}$ with constraint $x^2+4y^2 \le1$, Lagrange multipliers only help with finding the extreme values at the ...
0
votes
3answers
25 views

Functions - Inverses of graphs.

The question reads: sketch the graph of y=-3-x along with its inverse. From calculating the equation of the inverse graph, I come to y=-3-x, using the swap method. I then tried to plot both graphs ...
1
vote
1answer
23 views

Let $f\colon [a,b]\to\mathbb R$ is continuous and $G(x,t)=t(x-1)$ when $t\leq x$ and $x(t-1)$ when $t\geq x$.

Let $f\colon[a,b]\to \mathbb R$ is continuous and $$G(x,t)=\begin{cases}t(x-1)&\text{when $t\leq x$,}\\x(t-1)&\text{when $t\geq x$.}\end{cases}$$ Let $$g(x)=\int_0^1f(t)G(x,t)\,\mathrm dt.$$ ...
-2
votes
2answers
33 views

What is the function $w(x) = 4 + \sqrt[3]{x}$? [on hold]

I need to know how to go about determining if this function is even, odd, or neither
0
votes
2answers
49 views

Differentiability and continuity at the origin of piecewise defined $g(x,y) = y-x^2$, $y+x^2$, or $0$

$$g(x,y)= \begin{cases} y-x^2, & y\ge x^2\\ y+x^2, & y\le -x^2\\ 0 & \text-x^2\le y\le x^2 \end{cases}$$ I need to find all the directional derivatives at the origin in the tangent ...
2
votes
3answers
78 views

Number of functions verifying $f(f(x))=f(x)$.

Find the number of functions $f:\{1,2,3,4\}\to \{1,2,3,4\}$ that verify $f(f(x))=f(x)$. I'm not sure if the answer is $41$ or $29$.
4
votes
6answers
72 views

$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$

$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically? Using integration by parts I got the form: ...
1
vote
0answers
26 views

fn converges to f pointwise where all functions fn are bounded and f is unbounded: Is this example correct?

I am looking to find a sequence of functions $f_n$ that converges to a function $f$ pointwise, where all functions $f_n$ are bounded, but $f$ is unbounded. I have thought of an example where the ...
1
vote
1answer
10 views

What other types of distributivity are there?

When I say ‘Distributivity,’ I mean the way a number $x$ can be ‘Put in to’ some other function or the like. For example, to distribute $x$ into $\text{id}_y$, you simply have ...
4
votes
1answer
71 views

Solving the functional equation $f(x^2+f(y))=(f(x))^2+y$

Find all $ f : R\rightarrow R $ such that $f(x^2+f(y))=(f(x))^2+y, \forall\text{ x,y}\in R$ Thanks in advance!
0
votes
1answer
10 views

Writing a function in a part that is linearly dependent on the dependent variable

Let's say I have the function $f = f(x)$, Under what conditions am I able to write this function as follows: $f = c + h(x)x$ where $c$ is a constant, and $h(x)$ is some function depending on $x$.
0
votes
1answer
37 views

Evaluating $\lim_{n\to\infty}\int_0^1x^nf(x)\,dx$. [duplicate]

Let $f$ be a continuous function on [0,1]. Evaluate $$\lim_{n\to \infty} \int_0^1 x^nf(x)dx$$ My approach : Consider $\int x^nf(x)dx = \frac{f(x)x^{n+1}}{n+1} - \frac{1}{n+1}\int x^{n+1}f(x)dx$ ...
1
vote
1answer
72 views

Proof of Hyperbolic Functions

Find the proof:  (a) Use the definitions cosh(x)= 1/2(ex +e^−x) , sinh(x)= 1/2(e^x − e^−x) to express sinh(x + y) and cosh(x + y) in terms of cosh(x), sinh(x), cosh(y) and sinh(y). (b) Using the ...
1
vote
2answers
45 views

Can a real continuous bounded function on $ \Bbb{R}^{2} $ be expressed as a finite sum of products of real continuous functions on $ \Bbb{R} $?

Can a real-valued continuous bounded function on $ \Bbb{R}^{2} $ always be expressed as a finite sum of products of real-valued continuous functions on $ \Bbb{R} $?
-1
votes
0answers
24 views

Squares and Square Roots [on hold]

what are the laws/principles/rules to determine when to square or take a square root of a variable? To say it another way, how do you determine when you need to square or take a square root of a ...
5
votes
2answers
174 views

Why is $\sinh$ often pronounced “shine”?

I talked to some guys from the UK and they told me that they would pronounce $\sinh$ as "shine". I am not a native english speaker so I don't know, but in my country we call this function "sintsh" ...
0
votes
1answer
14 views

Precision needed in definition of unboundedness

This is a quick question about what it means for a function to be unbounded. Does it mean that the function tends to + or - infinity, or does it just mean that it has no limit?
1
vote
1answer
25 views

How can this function be considered to have a saddle node bifurcation?

Say I have the function $f(x,\mu) = (1 + \mu)x − x^2 − 0.1$. By definition a Saddle Node bifurcation occurs if: $f_{\mu_0}(0) = 0$ $f'_{\mu_0}(0) = 1$ $f''_{\mu_0}(0) \neq 0$ $\frac{\delta ...
1
vote
1answer
33 views

How to generate integer random numbers that equal to another random number?

I am running a simulation in Excel, and need to generate a group of integer random numbers summing up to another random integer, how can I possibly do it? For instance I have an integer random number ...
2
votes
2answers
36 views

Can we solve recurrence relations indexed by R?

Recurrence relations are equations with functions from $\mathbb N\rightarrow \mathbb N$. For example given $a:\mathbb N\rightarrow \mathbb N$ (write $a(n)$ as $a_n$) solving the recurrence relation ...
0
votes
1answer
36 views

e Function Parity

I have look at the plot of the function $$(1+\frac{1}{x})^x$$ Is it an odd or even function? it seems like one flip on the Y-axis and on flip on the X-axis but unlike odd it is a flip upward Aren't ...
1
vote
1answer
31 views

Some conditions on $\tilde f(x,y)=\begin{cases}\displaystyle g(x,y) & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$

The following function $$f(x,y)=\begin{cases}\displaystyle\frac{x^2 y^2}{x^2+y^2} & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$$ is differentiable in the origin and ...
-3
votes
1answer
24 views

Given that $f(x)=\frac{5}{x-8}$ and $g(x)=\frac{9}{x+9}$, find [on hold]

(a) $(f+g) (x)=$ (b) $(f–g) (x)=$ (c) $(fg) (x)=$ (d) $(f/g) (x)=$ I don't know how to calculate the result of (x) .
0
votes
1answer
59 views

If $f(U)=0$ then what is possible?

Let , $U=\left(0,\frac{1}{2}\right)\times \left(0,\frac{1}{2}\right)$ and $V=\left(-\frac{1}{2},0\right)\times \left(-\frac{1}{2},0\right)$ and $D$ be the open unit disk centered at origin of $\mathbb ...
0
votes
0answers
11 views

Shifting a series of functions whilst maintaining symmetry

I have a function y = a*Exp[-(x - b)^2/2*c^2] + d*Exp[-Abs[-e*x]] + f Which is symmetrical when the coefficient of b is equal to 0 however it loses symmetry as ...
0
votes
1answer
16 views

The Average Rate of Change of Function

Here is the question: The following chart shows the growth of a crowd at a rally over a 3 h period. Time (in hours): $ 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 $ Number of people: $0 , 176, 245, 388, 402, ...
-1
votes
0answers
22 views

Eliminate the arbitrary Function of PDE

I need to solve this problem; Eliminate the arbitrary function $f$ from the equation: $f(x^2+y^2+z^2,z^2-2xy)=0$ I try this solution $u= x^2+y^2+z^2, \quad $ $\quad v=z^2-2xy, \quad$ so ...