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

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0
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3answers
31 views

Determine the exact value of equations involving more two trig variables

$2\cos^2x=1+\sin x$. Determine the exact values of $x$ such that $0 \leq x \leq 2\pi$. I am experiencing problems with factoring this question. First I started by getting everything on to the ...
1
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2answers
21 views

Prove 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 such that $f'(x) > 0$ $\forall x \neq 0 \wedge f'(0) = 0 \wedge f''$ is a continuous one to one function on some open interval $(a, b): a < 0 < b$ then $f$ ...
0
votes
0answers
18 views

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$ I need to find the following: $(a)$ Show $F$ is one-to-one on $A$. $(b)$ Show that $F(A) = \{(u,v) : 0 < \frac{u}{2} < v ...
3
votes
0answers
47 views

How can we show that the functions are differentiable?

Show that the following functions $$f(x, y)=\frac{xy}{\sqrt{x^2+y^2}} \\ f(x, y)=\frac{x^2y}{x^4+y^2}$$ are differentiable at each point of the domain. Determine which of them is $C^1$. $$$$ The ...
1
vote
1answer
36 views

Determining the exact value of trigonometric functions using tan

Use the special triangles to give exact solutions where possible. Find all values of x such that $0 \le x \le 2\pi$. The question I have is $\tan^2x=1$. What I have done so far (it appears that ...
1
vote
1answer
24 views

Difficulty in understanding a piecewise function problem

I am using Stitz-Zeager PreCalculus book and I am not able to fully understand the problem. The Part I am feeling problem in is the c one. The Problem is as follows: original image For $n$ copies ...
3
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1answer
22 views

Special Triangles and their related acute angles

So I've been working on some questions involving having to find the exact value of trig. functions involving a particular interval. I have worked through the question but now I have something I am ...
0
votes
1answer
27 views

Find all values of $x$, linear and quadratic functions

Use special triangles to give exact solutions where possible. Find all values of $x$ such that $0 \leq x \leq 2 \pi$. 1) $\cos^2 x + \cos x - 1 = 0$ For this question, I have factored in which the ...
3
votes
0answers
33 views

How is $\cos^3{x}$ an odd function while $\sin^3{x}$ an even function?

We know that for odd function $f(-x) = -f(x)$ and for even function $f(-x) = f(x)$. Therefore, $\cos^3(-x) = \cos(-x)\cos(-x)\cos(-x) = \cos{x}\cos{x}\cos{x} = \cos^3{x}$ (i.e. $\cos^3{x}$ must be ...
0
votes
2answers
25 views

Inverse Image Proof

Let $f:X\rightarrow Y$. Let $A$, $A_1$ and $A_2$ be subsets of $X$ and $B$, $B_1$, and $B_2$ be subsets of $Y$. Then, I need to prove that $f^{-1}(B_1\cup B_2)=f^{-1}(B_1)\cup f^{-1}(B_2)$. I know ...
5
votes
1answer
47 views

Show that $f(a,b)$ is one-to-one

Let $$A=\{(x,y)\in\mathbb R^2:x>0, y>0\}$$ and define $f:A\to\mathbb R^2$ by $$f(a,b)=(a+b^2,2a^2+b).$$ Show that $f$ is one-to-one on $A$. I know that a function is one-to-one if all ...
3
votes
3answers
47 views

Prove $\lim_{x\to\infty} \left( \sqrt{x+1} - \sqrt{x} \right) = 0$

My attempt: I tried manipulating the formula, but I couldn't do anything useful. I tried to find another function $f(x)$ such that $\lim_{x\to\infty} f(x) = 0$ and $f(x) \geq \sqrt{x+1} - \sqrt{x} $ ...
0
votes
1answer
20 views

How to Prove this Fractional Linear Transformation of $\mathbb C$ takes $S^1$ to itself?

Let $x\in\mathbb C$. I know that $|x|<1$ but I don't think that matters for what I'm about to ask. Let $f$ be the fractional linear transformation $f(z)=\frac{z-x}{1-\overline x z}$. Then I'm ...
0
votes
0answers
29 views

Proof that set of functions with derivative zero at a given point is meager in space of strictly increasing twice differentiable functions.

Let $X = \{f: [0,1] \to \mathbb{R} \; | \; f\in C^2[0,1], f \textrm{ strictly increasing} \}$. I equip $X$ with the topology of uniform convergence. Define the set $A$ as: $$A =\{ f \in X \; | \; ...
0
votes
0answers
11 views

“Sigmoid” function with tunable initial slope, upper asymptote and transition period

I'm looking for a function which resembles the transition between the function $f(x)=x$ for small $x$ and the function $f(x)=C$ for large $x$ ($x$ is finite and $\geq 0$). I've found the generalized ...
-1
votes
2answers
32 views

How to find unknown variables?

Given that $f(x)= 8x+m$ and $g(x)= x^2 - 3x +n$ and $g\circ f(x) =64x^2-8x$ where $m $ and $n$ are constants. Find the values of $m$ and $n$. Can someone point me in the correct direction on how ...
0
votes
5answers
93 views

Range of $\cos(2 \sin x)$ [closed]

Actually, I am confused determining the range of the function given below. Could anyone tell me what the range of $ f(x) = \cos (2 \sin x)$ is ?
0
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0answers
22 views

Integral of a function depending on two variables

I have a function that depends on two parameters, say $f(x,y)$. Now I need to integrate this function from $0$ to $y_{max}$. I have all the values of $x$ and $y$ and also the values of this function ...
1
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4answers
48 views

Are all functions that have an inverse bijective functions?

To have an inverse, a function must be injective i.e one-one. Now, I believe the function must be surjective i.e. onto, to have an inverse, since if it is not surjective, the function's inverse's ...
1
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1answer
27 views

What is the name of the most locally convex / concave point of a $f(x)$ function?

I was looking for the name given to the more locally convex / concave points of a given function $f(x)$ for instance, the ones I have marked in the multiplicative inverse function below. In the case ...
4
votes
3answers
255 views

Transforming a function by a sequence geometric operations on its graph.

I am solving the following problem: Let $f(x) =\sqrt{x}$. Find a formula for a function $g$ whose graph is obtained from $f$ from the given sequence of transformations: shift right $3$ ...
0
votes
0answers
22 views

Expectation of function and its derivatives

I was wondering if anyone has an idea of what could be said about $f$ given that it satisfies the following inequality: $$\mathbb{E}\left[ \frac{1}{\sigma^2}f(x)^2+f'(x)^2+2f(x)f''(x) \right] \geq 0$$ ...
1
vote
2answers
66 views

Solution of recursive polynomial functions

Is there anything that can be said about the roots of the polynomial $f_n(x)$ if $f_n(x) = xf_{n-1}(x) + f_{n-2}(x)$ where these are polynomials of degree $n, n-1,$ and $n-2$, respectively? My goal is ...
0
votes
1answer
48 views

Question about necessary and sufficient conditions?

I am working on a question which begins with The number $\alpha$ is a common root of the equations $x^2+ax+b=0$ and $x^2+cx+d=0$. Given that $a\neq c$, show that $$\alpha=-\frac{b-d}{a-c}$$ ...
1
vote
1answer
32 views

Proving statements about ceiling and floor functions.

Prove or disprove the statements below. (a) For all positive real numbers x and y, $\lfloor x \cdot y\rfloor ≤ \lfloor x\rfloor \cdot \lfloor y\rfloor $. (b) For all positive real numbers x and y, ...
0
votes
1answer
18 views

Finding out how many distinct functions can be made.

The problem goes as follow: Let$ A = {1, 2, 3, 4, 5}.$ (a) How many total functions $f : A → A$ are there? (b) How many of the functions in (a) are one-to-one? I would say only one function can ...
0
votes
0answers
9 views

Alternative method for y-vertex calculation

So, I've been wondering the following: If you can determine the x coordinate of the vertex of a quadratic function by averaging the x coordinates of both roots, would it be possible to determine the ...
2
votes
1answer
36 views

How do I integrate the inequality $ \frac{f(\frac{1}{2}+h)+f(\frac{1}{2}-h)}{2} \leqslant f(\frac{1}{2})$ over the range $h\in[0,1/2]$?

I would like to know the formal steps and theory. I was told that, by integrating this inequality, I can achieve one of the definitions of a concave function in the interval [0,1]. Thanks for your ...
1
vote
0answers
17 views

Prove a lower bound $\left|\int_{-\infty}^\infty k(t) g(t) e^{i\theta t}dt\right|\geq C \int_{-\infty}^\infty |f(t)|dt$

Let $k(t)$ be any function absolutely integrable over $(-\infty,\infty)$ and let $$g(t)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(u) e^{itu}du$$ Consider $$\int_{-\infty}^\infty k(t) g(t) e^{i\theta t}dt$$ can ...
0
votes
2answers
38 views

What is the difference between domain and implied domain of a function?

I was solving exercises of pre calculus book when I came up with a question which asked me to find the implied domain. Please can anyone tell me? What's the exact difference between the two? I think ...
0
votes
1answer
21 views

Deriving a function based on a relation/characteristic

Say I give you an integer set [1, N], which is the initial step, and define a notion of a step by this example: given N=16 ...
0
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1answer
24 views

Let $f : R → R$. Sequence in image converges, prove convergence of the function at a given value in the sequence.

Let $f : R → R$. Assume $f$ is increasing. Assume $f(1) = 2$. Assume the sequence $2 + (−1)^n/n$ belongs to the image of $f$. Prove that $f$ is continuous at $1$. Should I just show the sequence ...
0
votes
0answers
15 views

True/False with regard to uniform continuity on sum of

I am having a lot of trouble proving or disproving these. Let $f_n : E → \Bbb R$ be continuous functions for $1 ≤ n ≤ N$. Let $a_k^{ (n)}$ be $N$ convergent sequences of numbers. Assume $lim_{k→∞} ...
1
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2answers
50 views

Using the Mean Value Theorem, show that if $f'(x) > 0$ $\forall x \in (a, b)$ then $f$ is increasing on $(a, b)$

Using the Mean Value Theorem, show that if $f'(x) > 0$ $\forall x \in (a, b)$ then $f$ is increasing on $(a, b)$. The Mean Value Theorem states: a function $f$ which is continuous on the closed ...
0
votes
3answers
52 views

If $f$ is a increasing function in $[a,b]$, then is it true that $\text{Img}(f) = [f(a),f(b)]$?

If $f$ is a increasing function in $[a,b]$, then is it true that $\text{Img}(f) = [f(a),f(b)]$? I am in doubt because my book said no. Is my proof correct? If $f$ is a increasing function, ...
1
vote
2answers
40 views

Prove that if $f$ is a function such that $f'(x) > 0$ for $x \in \mathbb{R}$ then $f$ is a one to one function.

Prove that if $f$ is a function such that $f'(x) > 0$ for $x \in \mathbb{R}$ then $f$ is a one to one function. Set $f(x)$ to be some function such that $f'(x) > 0 \implies$ f(x) is ...
1
vote
2answers
65 views

Optimization, find the dimensions of the poster with the smallest area

The top and bottom margins of a poster are 4 cm and the side margins are each 2 cm. If the area of printed material on the poster is fixed at 380 square centimeters, find the dimensions of the ...
0
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2answers
87 views

Prove whether or not the function is a bijection

Problem statement: Let $ f: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}$ be defined as $ f(m, n) = (3m + 7n, 2m + 5n) $. Is $f$ a bijection, i.e., one- to-one and onto? If yes ...
0
votes
1answer
33 views

Operations with functions

I am having issues with a functions problem. I have checked with other friends and they have gotten a different answer than me. I would like if you could check my work for me. I have done this problem ...
0
votes
1answer
36 views

Determining Exact Values of Trignometric Equations

Use the special triangles to give exact solutions where possible. Find all values of $x$ such that $0\le x \le 2\pi$ . (a) $\tan^2 x=1$ $\,$ (b)$\, \, 2\cos x + \sqrt{3}=0 \, \,$ (c) $\, \, ...
0
votes
2answers
57 views

Analytic functions with infinitely many zeros [closed]

What are some analytic functions, other than the common trigonometric functions, that have infinitely many zeros?
0
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1answer
9 views

How do I find the value in a range based on the value in another range?

For example Range1 = 0.6 to 0.7 Range2 = 0.0 to 0.5 When Range1 = 0.6(min) Range2 = 0.5(max) Inversely when Range1 = 0.7(max) Range2 = 0.0(min) So lets say the value of Range1 = 0.65, how do I ...
0
votes
0answers
40 views

Finding a function using first derivative

I have some data about just first derivative of a function. Also, I know a point of this function(e.g. (x1,y1)). How can I obtain the function? All my date are numerical. dev f(x)=[ 580.00 , 479.7308 ...
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2answers
40 views

Started this problem but can't finish it: Showing pointwise convergence for this summation

I know how to start this problem but am having trouble finishing the end of it. Any help would be great! Thanks We let $g_n: E \rightarrow \mathbb{R}$ be continuous functions for $1 \leq n \leq N$ ...
1
vote
1answer
51 views

Existence of functions $g$ such that 1. $f\circ g(1) =2$; 2. $g \circ f(1) = 2$, for all $f$ [closed]

Let $S = \{1,2,3,4\}$. Let $F$ be the sets of all functions from $S$ to $S$. a) Prove or disprove the statement: "For all $f \in F$, there exists $g \in F$ so that $(f \circ g)(1) = 2$" b) Prove or ...
1
vote
1answer
21 views

Absolute Maximum and Minimum of cos function

I am having a little trouble trying to figure out the following problem: Find the absolute maximum and minimum values of the function $f(x) = x-2\cos x$ on the interval $[0, 2\pi]$. I have taken the ...
0
votes
0answers
16 views

Find function $f(x,y)$ such that $f(x,0) = J(x)$ and $\nabla_{(x,y)} f(x,y) = g'(y)h(J,\nabla_x J)$?

Let $J:\Omega \to \mathbb{R}$ be a smooth function such that $0 < C_1 \leq J(x) \leq C_2 < \infty$. Is it possible to find a function $f:\Omega \times [0,\infty)$ such that $$f(x,0) = J(x)$$ ...
0
votes
1answer
43 views

real analysis -functional equation

Be $f:\mathbb R^ +\mapsto\mathbb R$ a function that satisfies the following conditions: a)$ f(f(f(x)))+2x=f(3x)$ for every $x\gt 0$; b) $\lim_{x \to \infty} (f(x)-x)=0$. This was proposed by ...
1
vote
1answer
26 views

Quasiconvex functions

Let $\lambda$ be any real number and $f:[0,1]\times[0,1]\rightarrow\mathbb{R}$ given by $$ f(x,y) = \begin{cases} \lambda &\mbox{if } \quad0<x<1, y=1, \\ 1+\lambda y & \mbox{if } ...
0
votes
1answer
19 views

Determine if a function is even or odd

Let $f:\mathbb{R}\to\mathbb{R}$. Define $h:\mathbb{R}\to\mathbb{R}$ by $$h(x)=f(x)\{f(x)+f(-x)\}$$ Then, which of the following option(s) is/are correct ? (A) h is even for all f (B) h is odd for ...