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

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1answer
30 views

Periodic functions which is continuous [on hold]

$f$ is a function from $\mathbb R$ to $\mathbb R$ with period $T > 0$. Prove that there exists a $c$ such that $f(c)=f(c+ \pi)$
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votes
2answers
24 views

Show that there exist $x_{0}\in[0,1]$ such that $f(x_{0})=g(x_{0})$ [on hold]

Let $f,g:[0,1]\rightarrow[0,\infty)$ be continuous such that $\smash{\displaystyle\max_{x \in [0,1]}} f(x) = \smash{\displaystyle\max_{x \in [0,1]}} g(x)$. Show that there exist $x_{0}\in[0,1]$ ...
0
votes
1answer
10 views

primitive recursive conditional

I am confronted with the assertion that the following expression describes the conditional: $\text{Cond}\left[ t, f, g \right] = \text{Pr} \left[pr^2_1, pr^4_2 \right] \circ(f,g,t)$. This is meant ...
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votes
5answers
59 views

Solving equations.

How would you solve these equations and show that they do not intersect each other? $$x^2+y^2=2x-2y$$ $$x^2+y^2=4(x^2+y^2)^{1/2} +y$$ It's isolating a term which I am struggling with. General ...
0
votes
1answer
23 views

Uniform and pointwise convergence

So we had the pointwise and uniform convergence, and I do get that a sequence of function can converge to a function, just like ordinary sequences do. But what I don't quite get is this pointwise and ...
-1
votes
1answer
13 views

Linear functions and equations in text

The question is: A rectangle has a width of 28cm. The one side is 2cm longer than the other side. We are supposed to form an equation and then solve it, but I don't know to form an equation ...
2
votes
2answers
45 views

The Cantor staircase function and related things

The Cantor staircase function https://en.wikipedia.org/wiki/Cantor_function has an interesting property: $\{x\colon f'(x)\neq 0\}$ is a nowheredense nullset. But it it differentiable almost ...
-2
votes
0answers
43 views

Is this an incorrect way to define a function?

This question spawned from this MathSE thread. Define a function $f:D\rightarrow R$. ($D$ for domain and $R$ for range.) I want to "approximate" $f$ by a function $g$, but I only want to define $g$ ...
0
votes
0answers
28 views

Do partial derivatives determine function uniquely?

When we talk about functions of one variable, we have less complicated situations compared to the case in functions of several variables. In the one variable case, for continuous functions ...
0
votes
1answer
44 views

Solution of $(x^2+f^2(x))f'(x)=1,\forall x\in [1,+\infty)$

A question of interest that arose during my doing a seemingly easy exercise is: The exercise considered a function $f:[1,+\infty)\to\mathbb{R}$ differentiable with $f(1)=1$ and $f'(x)\left( ...
4
votes
1answer
30 views

Length of wire needed to graph a function

Question: $$S = {(x,y): \left|||x|-2|-1\right| + \left|||y|-2|-1\right| = 1}$$ If S is made out of wire, find the length of the wire required. I have no clue as to where to begin this ...
0
votes
1answer
41 views

Prove or disprove: If $1=||A||>||B||$, then $A-B$ is nonsingular.

Prove or disprove: If $1=\|A\|>\|B\|$, then $A-B$ is nonsingular. I think that since $\|A-B\|>0$ by the given conditions we know it is nonsingular. Any solutions or hints are greatly ...
0
votes
2answers
31 views

What's the point of an inclusion function defined $\to X$?

The inclusion function is defined in my notes as follows Let $A \subseteq X$ for any set $X$. The inclusion function $i:A \to X$ is defined by $i(a)=a$ $\forall a \in A$. Well, what I don't get ...
0
votes
3answers
33 views

Proving a Function is continuous on an interval.

For the function $f(x) = \frac {1}{\sqrt{x}}$ Show the function is continuous on (0, $\infty$) How do I approach/do this question?
0
votes
0answers
26 views

Propertie of Convex and concave function $f(x)\text{ bigger than }f(f(x))$

Is it true that if $f:\mathbb{R} \to \mathbb{R}$ is convex or concave in some interval $[a,b]$ then $f(x)<f(f(x))$ or $f(x)>f(f(x))$ respectively.
1
vote
1answer
38 views

Let $α$ and $β$ be the roots of equation $px^2+qx+r=0,p≠0$

Let $α$ and $β$ be the roots of equation $px^2+qx+r=0,p≠0$, If $p,q,r$ are in A.P and $\dfrac{1}{α}+\dfrac{1}{β}=4$, then the value of $|α−β|$ is $:$ $\dfrac{\sqrt{61}}{9} $ $\dfrac{2\sqrt{17}}{9}$ ...
0
votes
2answers
36 views

A better general definition of a predicate

What's a better definition for (an interpretation of) a predicate in general (i.e. non-theory-specifically): ...
0
votes
2answers
27 views

What will be the domain of this function?

Question: find the domain of the following function $g(x) = (x^2 - 6x)^{1/4}$ My try: Since the expression $x^2 - 6x$ is under an even root, then the expression should be greater than or equal to ...
0
votes
0answers
19 views

Lagrange function on MATLAB

I'm trying to write the Lagrange function in Matlab and I need some help. This is what a friend and I have got so far, I am just not getting how to finish: ...
0
votes
0answers
44 views

Find the points on the curve $y=x+e^x$ at which the tangent line is horizontal.

Find the points on the curve $y=x+e^x$ at which the tangent line is horizontal. The answer was $(0,1)$, but I don't get it. I tried to take the derivative of the function and equal it to $0$ ...
2
votes
1answer
13 views

Can I check whether it is correct about restriction of a function.

I've not learned about restriction of a function. However, the solution in the web-site is using the restriction of a function. Thus, I read its definition in the wikipedia. However, I do not sure ...
4
votes
1answer
18 views

Let $f:A\to B$ and $g:B\to C$. Suppose $g\circ f$ is a bijection. Then $f$ is injective and $g$ is surjective onto $C$.

I think proving $f$ is injective is fairly simple: Let $x_1,x_2\in A$ s.t. $f(x_1)=f(x_2)$. Then, $g\circ f(x_1)=g\circ f(x_2)$. Thus, as $g\circ f$ is bijective, we have that $x_1=x_2$. Thus, $f$ is ...
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0answers
22 views

Graph of a higher-order function

When we deal with functions which work on numbers, we can graph them easily: Just take each of its possible input values and find its corresponding point on one axis, then go straight up in its ...
1
vote
3answers
34 views

Finding set of values of P for which f(x)=p has no real roots

I'm not much into Algebra and Functions so I got confused in this one. The question ( I'll write it exactly how it asks ) says, The function f is defined by $f : x  \to 2x^2 -6x + 5$ for $x \in ...
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vote
2answers
73 views

Integration equals another integration

$\displaystyle\int f(x)\ dx=\displaystyle\int g(x)\ dx$ So what is the relation between $f$ and $g$ I found this solution but i am not sure it is right or not : $\displaystyle\int (f(x)-g(x))\ dx=0$. ...
0
votes
1answer
23 views

Find the number of local Maxima in the function.

question is simple: Find the number of local maxima in the following function $$f(x)=\cos (2\pi x )+x-\{x\}$$:where {...} represents fractional part of $x$, in the interval $[0,10]$ I plotted ...
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vote
0answers
19 views

A property of ratio of two polynomials

I am given a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x)= \frac{p(x)}{q(x)}$ for all x, and $p(x),q(x)$ are two polynomials with integer coefficients. I have two questions related to ...
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0answers
11 views

periodic function sine wave problem relate with sound

SinD +SinC = 2Sin (C+D)/2 Cos(C-D)/2 can anyone help me on how to express this equation as single wave in the case that two sine waves those sum makes a sound? Thank You!
2
votes
0answers
46 views

Maxima and minima of $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}$.

Find the maximum and minimum value of the function $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}.$ ATTEMPT:- By A.M.-G.M. inequality, $\frac{a+b}{2}\ge\sqrt{ab}$, $\quad$ for $a,b\gt 0$ with equality ...
1
vote
1answer
44 views

Smooth round hills with an off-center peak

I want smooth roundish hills, functions $f: \mathbb{D} \to \mathbb{R}$ on the unit disk, with a single peak at a given point $p$, e.g. $p = 0.5$ tapering smoothly to $0$ on the boundary, $|z| = 1$. ...
0
votes
1answer
10 views

Why is the composition of either an odd or an even function with an even function even, but not vice versa?

This is a statement from wikipedia. I don't understand what the "but not vice versa" part means. Does it mean that the composition of even or odd function with even is not odd (can only be even)? Or ...
0
votes
0answers
24 views

Convergence of a function of two variables

The following question has been posed to me by a student in an analysis class. For which real numbers $\alpha \gt 0$ is the function $f : \Bbb R^2 \to \Bbb R$ given by $f(x, y) = (x^2 + y^2)^\alpha$ ...
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vote
4answers
45 views

Mathematical Notation of Sequence of Functions

Let's say I have a finite set of functions $F=\{f_1,f_2,f_3,...,f_n\}$ and I want to show a recursive function that is constructed by an arbitrary sequence of applications of functions in $F$ to input ...
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2answers
26 views

Determine if the following composition function is onto

Define $f: \Bbb{Z}\times\Bbb{Z} \to \Bbb{Z}\times\Bbb{Z}$ and $g: \Bbb{Z}\times\Bbb{Z}\to \Bbb{Z}\times\Bbb{Z}$ by: $$f((a,b))=(a+b,2a)$$ $$g((c,d))=(c+2d,c)$$ Determine if $g \circ f$ is onto. I ...
1
vote
1answer
23 views

Understanding a detail about functions

Let $\sigma, \tau, \pi$ be functions such that the following compositions make sense. Assume the following is true: $\sigma$ is surjective(not sure this is needed here), and for $\sigma$ and $\tau$ ...
0
votes
1answer
21 views

How to shift the weighted mean of a monotically increasing series of values

I have a monotonically increasing series of values $X, x\in[0,1] \ \forall i\in1,2,...,60$ the weighted mean of which is defined by $$\bar{x}=\frac{x_1+\sum_{i=2}^{60} i(x_i-x_{i-1})}{x_{60}}.$$ Given ...
0
votes
0answers
39 views

Equality of two functions.

I have a specific question, from a paper given below. Here I got an answer of question: When two functions are called equivalent?.It helped me to understand the first and the second steps of the ...
0
votes
2answers
24 views

Differentiable function and increasing and decreasing problem

I dont get why (I) is not correct. The slope at 3 is positive, meaning the f'(x)>0. Can anyone explain? Correct answer is C
4
votes
2answers
57 views

What do uniformly continuous functions look like?

When I see a function, I want to be able to quickly determine whether it is uniformly continuous or not. Usually, this kind of skill comes after being exposed to many different examples that either do ...
1
vote
2answers
28 views

Implicitly finding the derivative of $f^{-1}(x)$ given $f(x)$

Can we find the derivative of the inverse of a function implicitly by finding the derivative of the original function? For example lets say I have $f(x) = e^x$ and I want to find the derivative of ...
0
votes
4answers
39 views

A question involving logarithms; How to solve? [on hold]

Consider the function $f$ defined by $f(x)=\ln(\ln x)$ Questions: Prove without use of derivatives, that $f$ is strictly increasing. Show that there exist an unique real number $x$, ...
0
votes
1answer
8 views

Clarification on the idea of absolute maxima

$$f(x)=-|x|\:,\:\:x≠0$$ If f(x) did not have an point of discontinuity at x = 0, then it is obvious it would have an absolute maximum there. However, now that that point no longer exists, does it ...
0
votes
1answer
41 views

Sketching functions.

1) The functions $f$ and $g$ are as follows. $$f(x) = x^2 + 4x,\; x≥-2$$ $$g(x) = x + 6 ,\;x\in \mathbb R$$ i) Show that the equation $g\circ f(x)= 0$ has no real roots. $g\circ f(x)=(x^2 + 4x) + 6 ...
1
vote
1answer
21 views

what is the maximum value of $f(x,y)=100x-x/4y-ln(y)-\frac{1}{2}x^{2}$ [on hold]

I want to know the maximum value of the function $f(x,y)=100x-x/4y-ln(y)-\frac{1}{2}x^{2}$, in which $x\geq 0$ and $y>0$. Can anybody help to find out the maximum value? Thanks
0
votes
1answer
34 views

How to count even numbers in the first n natural numbers?

So let's say that I count $1,2,3,4$ and ask the question: How many even numbers are there in this sequence. Well, there are $2$ even numbers or we can say $4/2$ even numbers. But if the count is ...
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votes
0answers
18 views

Using integration to find volume of a parabolic prism.

Suppose I have a solid with its profile being a parabola given by the equation $y = f(x)$. Its depth at any point is given by $g(x)$ and its width is a constant $k$. This essentially results in a ...
0
votes
1answer
29 views

If $f$ and $g$ are invertible and $f\circ g$ is defined, is $g^{-1} \circ f^{-1}$ defined?

In the proof for that any invertible functions $f$ and $g$ with $f \circ g$ defined, $(f\circ g)^{-1} = g^{-1}\circ f^{-1}$, it seems to me that there is an assumption that $g^{-1}\circ f^{-1}$ is ...
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votes
2answers
34 views

Functions Solving Equations (URGENT) (EDITED) [on hold]

Edit: NOTE: All the questions are linked. I.E the function for $a$ may be in the first question etc I'm really having issues with this so your assistance is very much appreciated. I am having ...
0
votes
1answer
54 views

Is that true that not every function $f(x,y)$ can be writen as $h(x) g(y)$?

If not, why? Here $h$ and $g$ are two general function.