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

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0
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0answers
13 views

Does $f ^{-1}( x )$ continuous if $f(x)$ is a continuous function and $f'(x)>0$

If $f(x): [a,b] \to \mathbb{R}$ is a continuous function and $f'(x)>0 \ \forall x \in (a,b)$. Dose $f^{-1}(x)$ continuos ?
0
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0answers
8 views

if $f:X\to Y$ is a surjective, then there is an injective function $g:Y\to X$ [duplicate]

I'm trying to prove if $f:X\to Y$ is a surjective, then there is an injective function $g:Y\to X$. I know this intuitively, since $f$ is surjective, for every $y$ in $Y$ we need just to choice an ...
5
votes
1answer
92 views

What does it mean to extend a function?

What does it mean to extend a function? Can someone please give an example? Thanks in advance!
0
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0answers
7 views

Looking for a “standard-mollifier”-type function

I'm looking for a function $\phi$ that behaves like a standard mollifier, in particular it should be positive, even, $\phi(0)=1$, $\phi(x)=0$ for $|x|>1$ and $C^\infty$. The (scaled) mollifier ...
2
votes
2answers
52 views

Show that an inverse mapping $ g = f^{-1}: J \to I $ is continuous

Question: Let $f: I \to \mathbb R $ be a continuous, injective funtion. Show that its inverse mapping $ g = f^{-1}: J \to I $ is continuous, where, $I = (x,y)$ and $J = f(I)$. I understand this ...
3
votes
3answers
34 views

How can I prove that a continuous injective function is increasing/decreasing?

If I have a continuous, injective function mapping the real numbers, then it is either increasing or decreasing. This seems intuitively obvious but I can't come up with a neat proof for it.
-1
votes
3answers
22 views

proof of an differential of absolute value of a function

Suppose that $f:\mathbb R \to \mathbb R$ is differentiable at $c$ and that $f(c)=0$ , show that $$g(x)=|f(x)|$$ is differentiable at $c$ if and only if $f′(c)=0$.
4
votes
3answers
96 views

Show that $f(x) = \frac{x^3}{1+x^2}$ is bijective

This seems like a simple question, but I'm stuck: how do I show that $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as $f(x) = \frac{x^3}{1+x^2}$ is bijective? I want to demonstrate that it is both ...
0
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0answers
18 views

Find monotony of $f(x)=x^5 +x^4 -11x^3 + 9x^2$

How can I find the monotony of $f(x)=x^5 +x^4 -11x^3 + 9x^2$ ? Derivative doesnt help much. Is there any other approach ?
1
vote
1answer
35 views

Series convergence or divergence how to test

I have the following series defined. $$\displaystyle\sum_{k=1}^{n} \cos \left( {\frac{\pi}{2}} k \right) \frac{k}{k+1000} \frac{1}{\sqrt{k}}$$ where $n = 1,2...$ How to test whether this series ...
1
vote
2answers
34 views

Show function g(x) is continous

I'm not able to understand how to prove this theorem. Consider a continuous function, $f : \mathbb{R} \rightarrow \mathbb R$. Using the definition of continuity, show that the function $g(x) = ...
0
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1answer
35 views

Prove that $\int_0^{\pi} \sin^nx\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cos(n+2)xdx=0$

Prove that $$\int_0^{\pi} \sin^nx\cdot\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cdot\cos(n+2)xdx=0$$ with $n \in \mathbb{N}$ I think it's true, but I can't prove.
0
votes
2answers
10 views

number of one-one function;a set to itself

How do you find the number of all one-one function from a set to itself? If you are asked to find the number of all one-one functions possible from any set A to itself ,how do you do it?The following ...
-3
votes
1answer
21 views

Need help with one of my homework questions [on hold]

Express the function in the form $f(g(x))$ $F(x) = (x^2 + 1)^{10}$
1
vote
1answer
35 views

Equality on functions in $ \mathbb{R}^n $

Let $ f,g : M \subset \mathbb{R}^p \to \mathbb{R}^q $ continuous. Given $ a \in M $, supose that all open ball centered in $a$ contains a point $x$ such as $f(x) = g(x) $. Show that $ f(a) = g(a) $. ...
0
votes
0answers
21 views

Does the range of all square root functions always equals to 0?

I know that the domain of the square root function has to equal to the minimum possible number (0) to have all the other numbers equal to 0 or be more than 0, but since the x value has to make the ...
0
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2answers
35 views

Inverse a function

I have problem to inverse this function , Can anyone help me to solve it?
1
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0answers
34 views

Bijection from $\mathbb{R}^2$ to $\mathbb{C}$.

Is there a frequently used letter to denote the obvious bijection $(x,y)\mapsto x+iy$ from $\mathbb{R}^2$ to $\mathbb{C}$? Cheers!
0
votes
1answer
39 views

Proving that $f(x)=\frac{1000x^{14}-7x^{11}+12x+7}{(x^7-1)^2+1}$ is a bounded function

I want to prove that function $f: \Bbb{R} \rightarrow \Bbb{R}$ such that: $$f(x)=\frac{1000x^{14}-7x^{11}+12x+7}{(x^7-1)^2+1}$$ is bounded. A good place to start would be to check limits as x goes ...
0
votes
3answers
24 views

Proof that the graph of a linear function and its inverse cannot be perpendicular.

I am refreshing my high school maths and got an exercise to proof that the graph of a linear function and its inverse cannot be perpendicular. Below is my proof. A linear function is a straight ...
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0answers
11 views

How to draw this domain?

I have to draw this domain defined by : For $\alpha \in [0,\frac{\pi}{2}[$, $D_\alpha := \{z\in \mathbb{C} / \vert z \vert <1$ and $\exists \rho \in ]0,cos(\alpha)], \exists \theta \in ...
1
vote
1answer
38 views

Splitting polygon in half. [on hold]

Let $P$ be a convex polygon in the plane. Prove that there is a vertical line which splits P onto two polygons of equal area. I tried to use intermediate value theorem with no luck.
0
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0answers
27 views

show that the function below is linear [duplicate]

let $f$ be a continuous function from $R$ into $R$ with this property: $f(x+y) = f(x) + f(y)$, for all $x,y \in R$. Prove that $f$ is linear.
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1answer
39 views

Solving for this function

From the Indian National Mathematics Olympiad 1992: Determine all functions $f: \mathbb R -[0,1] \rightarrow \mathbb R$, satisfying the functional relation: $$f(x) + f \left( \frac{1}{1-x} \right) ...
2
votes
2answers
161 views

Solving a second-degree exponential equation with logarithms

The following equation is given: $8^{2x} + 8^{x} - 20 = 0$ The objective is to solve for $x$ in terms of the natural logarithm $ln$. I approach as follows: $\log_8{(8^{2x})} = \log_8{(-8^{x} + ...
0
votes
1answer
18 views

Laplace using $t$-shift

Hello I'm doing a exercise that says to use the $t$-shift to find inverse laplace of $$ F(s) = \frac{5s+4}{s^2} \cdot e^{-2s}. $$ I'm not sure how to rewrite it so they have the same shift term on ...
0
votes
1answer
15 views

return a lower number based on a higher base number and vice versa.

So I am making an attack timeout based on a speed rating for a program that I am writing, but I am no genius when it comes to math. As it is i have a formula like so ...
1
vote
1answer
40 views

Connectedness, continuous functions, and the intermediate value theorem

I want to prove that for a continuous function mapping a connected space to ℝ such that f(p) never equals s, it follows that f(p) < s for all p or f(p) > s for all s. So here's what I know so ...
0
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0answers
13 views

Upper-bounds to $B_z(a,b)$

Is there any standard technique to produce nice upper-bounds to the incomplete beta function $$B_z(a,b)=\int_0^z t^{a-1} (1-t)^{b-1} dt \,?$$ Disclaimer: this question is intentionally not too ...
0
votes
1answer
23 views

Big-O of a Function

Given $F(N) = 55N(600 + 50N \log N + 20N) + 20N(30N + 20\sqrt N)(50 + \log N)$. How can one combine multiplication with addition for a Big-O estimate by algebraic means? I'm used to simply taking the ...
1
vote
1answer
11 views

Projective transformation

I need to find the function $f$ that satisfies the following: $f((1:1:0))=(0:1:1)$ $f((0:1:1))=(1:0:1)$ $f((1:0:1))=(1:1:0)$ If I let: $x=(1:1:0)$ $y=(0:1:1)$ $z=(1:0:1)$, then I get $f(x)=y$, ...
1
vote
1answer
24 views

Find a function that matches the following conditions.

Find a function that matches the following conditions. (a) $f(x)$ is continuous for all real numbers (b) $f(0)$ = 3 (c) For all real numbers $x$, $f(x) = f(x/2)$ This is from a past paper, and the ...
0
votes
3answers
41 views

Find the period of $|\sin x| + |\cos x - 1|$

I want to find the period of this function , I know that the period of $|\sin x| + |\cos x|$ is $π/2$ but what can a $-1$ do?
-1
votes
0answers
36 views

Homework problem involving composing functions [on hold]

$$f(x)=\log_3(2x-3),\;g(x)=5^x \implies f\circ g-1(25)=?$$ Please I want the steps of the solution and thanks for your help.
-1
votes
3answers
27 views

Is $c$ (as in $y=mx+c$) the $x$ or $y$ intercept?

In $y=5x+6$, is the $6$ the $y$-intercept or the $x$-intercept? I can't remember and need to know for revision.
-1
votes
1answer
31 views

Find the total number of functions. [on hold]

Consider the two sets $A=\{1,2,3\}$ and $B=\{1,2,3,4,5\}$. Then find the total number of functions from $A$ to $B$ and also find total number of one to one functions from $A$ to $B$.
0
votes
1answer
10 views

A limited composition of two unlimited functions on natural numbers?

Can someone give an example of two functions $f,g:\Bbb N\to \Bbb N$ such that $|\operatorname{Im}f|,|\operatorname{Im}\,g|\notin\Bbb N$, but such that $|\operatorname{Im}\,g\circ f|\in\Bbb N$?
2
votes
3answers
29 views

Is there a function that can reproduce this simple pattern?

input: 0, 1, 2, 3, 0, 1, 2, 3... output: 0, 1, 1, 0, 0, 1, 1, 0... I'm trying to resolve a function that takes the input and produces the corresponding output ...
20
votes
7answers
2k views

How to make a “function”?

I dropped out of school early when I was still a teenager and now I'm trying to take my GED. I'm really close to passing but I'm still having trouble understanding some concepts. In the pre-test, ...
0
votes
1answer
10 views

Range, Domain, Inverse of a Function

Define a function on the real numbers by: $f(x)=\frac{x+1}{x-1}$ and calculate the following: $\text{Dom }f=(-\infty, 1)\cup(1,\infty)$ $\text{Range }f=\text{Dom ...
8
votes
1answer
80 views

How many expressions can be formed with two commutative and associative functions?

EDIT: I have posted a generalization of this question to MathOverflow here. Suppose we have two binary functions $f,g$ which are commutative and associative, i.e., satisfying $$ f(a,b) = f(b,a) ...
3
votes
1answer
25 views

Problem with the type of equation $\sqrt{x}+\sqrt{y} = \sqrt{a}$ and vertices?

I am asked to find the type equation $\sqrt{x}+\sqrt{y} = \sqrt{a}$ , represents ? i.e a parabola , or hyperbola or ellipse or circle by squaring twice? Now , what I have done is like this ...
0
votes
1answer
43 views

Can we claim that $f(x)$ is an increasing function and its stationary point is $x = \infty$ (in this specific case)?

If $f(x)$ is concave and its first derivative $f'(x) \rightarrow 0$ when $x\rightarrow \infty$, can we claim that $f(x)$ is an increasing function and its (only) stationary point is $x = \infty$ ? ( ...
0
votes
1answer
16 views

Shifting Curves: Is this the correct way of doing it?

Say I am given a sketch of some function $y=f(x)$, and were to draw $y=f(2x-3)$ based on the given diagram. Would I be correct in saying this? Shift the curve to the right by 3 units, thereby ...
0
votes
0answers
13 views

What is a parametrized function?

I am readimg this article: Stochastic Gradient Descent Tricks and I would like some precisions: Each example $z$ is a pair $(x, y)$ composed of an arbitrary input $x$ and a scalar output $y$. We ...
0
votes
0answers
17 views

Can this multivariable function exist?

(3) Is there a function of two variables whose z = 0 level curve consists exactly of the circles $x^2$ + $y^2$ = 4 and $x^2$ + $y^2$ = 10? If so, what is an example? If not, why not? I initially ...
0
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0answers
25 views

Can't Understand Graphing this Function

I'm graphing a function, $1/2(4-2x)^{1/2} + 1$, and I'm establishing the following: My parent function: $x^{1/2}$ $(x, f(x))\longrightarrow(-(1/2)x - 4, (1/2)y + 1)$ I don't understand why when I ...
2
votes
1answer
47 views

Equivalent of $\int_0^{\pi/2}\cos^n(\sin(x))dx$

Let $\displaystyle u_n=\int_0^{\pi/2}\cos^n(\sin(x))dx$. How can I find an equivalent of $u_n$ when $n\to\infty$ ?
1
vote
1answer
52 views

Equivalent of the sum $\sum_{n=1}^\infty\frac{x^n}{\sqrt{n}}$

Let's consider $\displaystyle f(x)=\sum_{n=1}^\infty\frac{x^n}{\sqrt{n}}$. Where $f$ is defined, can we find a closed form for $f(x)$ ? What would be an equivalent of $f$ near $1^-$ ?
0
votes
1answer
27 views

Sense of the graph of a function

What makes it necessary to define the graph of a function $f:A\rightarrow B$ as $$\{(x,f(x))\mid x\in A\}$$ which makes it a subset of $A\times B$, when this is equal to the function itself, which is ...