# Tagged Questions

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

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### Differentiation and 3 dimensional

Assume that a Mountain is shaped like a function $z=f(x,y)=sin(xy)$. A Hiker begins at the Point $(0,0,0)$ and wants to reach the Point (1,1,sin(1)), but he is not allowed to get over the Slope of ...
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### Relations and restriction of a function.

This is a homework question: "Let R be an equivalence relation on a set S. For A ⊆ S, we define RA to be the restriction of R to elements of the set A, i.e., RA is a relation on A such that for any ...
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### How to see a function is $C^2$

Let $\phi_{z}(w)=\frac{z-w}{1-w\bar{z}}$ which is a conformal mapping of $\mathbb{D}$. $f\in C(\bar{\mathbb{D}})\cap C^2(\mathbb{D})$. $\mathbb{D}$ is the unit disk centered at the origin in the ...
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### Determine if $f=\{(x,y)\mid 2x+3y=7\}$ is invertible. From $\mathbb R \rightarrow \mathbb R$. If it is invert it.

I am thinking this is no, because I am not even sure if this counts as a function? I am unsure how this can be a function if there exist only a few $(x,y)$s that fulfill the equation. Or does the ...
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### “Inverse” of a step function

How can I write the function below $$f(x)=\left\{\begin{array}{ll} 1, & 0\leq t\leq 1, \\ 0, & t>1 \end{array}\right.$$ using the unit step function? I mean, I don't know how could I write ...
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### Proving h(x) = f(x)+g(x) is one-to-one where f and g are one-to-one functions

I have attempted to solve the question below, but I am not sure if it is correct. Let S = {1,2,3,4} and let F be the set of all functions from S to S. Let R be the relation on F defined by: For all ...
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### Need hint on differentiable exercise [closed]

I'm looking for a sequence of continuously differentiable $f_n : [0,1] \to \mathbb R$ functions, such that $f_n \rightarrow 0$ uniformly, but $f'_n$ doesn't converge uniformly. Could someone give me ...
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### How to prove (given $f\colon A\to B$ and $g\colon B \to C$), if $g∘ f\colon A \to C$ is onto, then $g$ is onto; if $g∘ f$ is 1to1, then $f$ is 1to1

I am doing practice problems for my exam, and I can't seem to figure this one out. Let 𝑓: 𝐴 → 𝐵, 𝑔:𝐵 → 𝐶. Prove that: (a) if 𝑔 ∘ 𝑓: 𝐴 → 𝐶 is onto, then 𝑔 is onto (b) if 𝑔 ∘ 𝑓: 𝐴 → 𝐶 ...
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### Proving that $f$ is the identity function if $\ f \circ g = g \circ f$ for all $g$.

This is a problem from Spivak's Calculus. I can see how it applies for some family of functions, for example, if $g$ is constant; but I need to prove it for any $g$.
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### Correct notation with composite function & characteristic functions.

I have the functions $p: \mathbb R \to \mathbb R; \quad p(x) = \frac12 x + 1$ $q: \mathbb Z \to \{0, 1\}; \quad q(x) = \begin{cases} 1 & x \geq 1 \\ 0 & x \leq 0 \end{cases}$ I know that ...
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### Dependence and Independence of $\epsilon$ and $\delta$

This is a question regarding Epsilon-Delta proofs, in this example in Single-Variable Calculus, but hopefully the crux of what I'm asking here, will be general enough. Introduction I'm putting ...
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### In the function $r :\mathbb{R} − \{0\} \to \mathbb{R}$, what effect does the “$-\{0\}$” have on the domain?

The full function is: $r : \mathbb{R} − \{0\} \to \mathbb{R}$ defined by $r(x) = 6 / x$.
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### Inverse image of function

Let $f:A\to B$ be a function such that $A= \{1,2,3\}, B=\{1,2,3\}$ $f(1)=1 , f(2)=1, f(3)=2$ Then is the inverse image of $f^{-1}(B) = \{1,2,3\}$? And, $f^{-1}(1) = \{1, 2\}$ but is it okay ...
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### Can a function have a set as argument or value?

I was joking with someone when I spelled some proper noun in a mathematical equation: $$\tan x \in f(u)$$ Obviously this function needs to have a value that is a set in order to make sense. Hence, ...
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### how to find mininimum $f(x)$ using $\int_{-\infty}^{\infty} f(x)g(x)dx$?

I would like to know the $f(x)$ which minimizes the $\displaystyle\int_{-\infty}^{\infty} f(x)g(x)\,dx$. Actually, this question start from the MMSE (Minimize Mean square error) ...
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### Proving the discontinuity of $g(x)=\sqrt{2+\tan^2x}$ without the aid of a graph

How would you show where the discontinuity of $g(x)=\sqrt{2+\tan^2x}$ is without a graph? What sort of approach would you take mathematically? For example, using the method for piece-wise functions. ...
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### Let $f(x)$ be an increasing function. Assume its image $f(C)$ is also a connected set. Prove that $f$ must be continuous

Let $f: R → R$ be an increasing function. Assume that for every connected subset $A$ of $R$, its image $f(A)$ is also a connected set. Prove that $f$ must be continuous. To prove this, I am thinking ...
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### Prove if a continuous function $f$ is one-to-one, it is monotonic.

This is the converse of Prove that if function f is monotonic, then it one-to-one. Let $f:(a,b) \mapsto \mathbb{R}$ be a continuous function. Prove if $f$ is continuous in $(a,b)$, $f$ is monotonic. ...
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### What is $\lim\limits_{n\to {\infty}} (\frac{n}{1+n})^n$? [duplicate]

What is $\lim\limits_{n\to {\infty}} (\frac{n}{1+n})^n$. Is it possible to write the function $f(x)=x^n$ and since we know $\frac{n}{1+n}\to 1$, so $f(\frac{n}{1+n})\to 1^n=1$. So the limit it $1$. ...
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### Having trouble understanding how to disprove/prove if a formula is a function.

Is $\frac 1{x^2-2}$ a function from $\mathbb{R}\to \mathbb{R}$? Is it a function from $\mathbb{Z}\to \mathbb{R}$? I have been thinking about this but, I can't find any example for which you can have ...
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### Time taken to run a function R(n,a)

R(n, a){ if n = 1 return(a); if n > 1 return (R(n − 1) + R(n − 1) + 1); } Could you please explain me why the estimated time taken to run R(n, a) as a function of n is: (2^(n−1))*(a + 1) − 1 ? ...
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### Is this true :if $x\in [0.2]$ then $f(x)=\frac{2x+3}{x+2} \in [0.2]$?

I'm sorry to ask this question mayeb it's a trivial question but i would like to confirme if i have this function $f(x)=\frac{2x+3}{x+2}$ which $x$ is a real number in $[0.2]$ then $f(x) \in [0.2]$ ...
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### Let $f:A\to B$. For each $B\subset T$, we have $f[f^{-1}[B]]=B$ iff $B\subset range(f)$

Let $f:S\to T$. For each $B\subset T$, we have $f[f^{-1}[B]]=B$ iff $B\subset range(f)$ I have the following to prove the $\to$ of the iff: Let $B\subset T$. Suppose $f[f^{-1}[B]]=B$. *Then, ...
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### What can I infer about $h(x,y)$ and $g(z)$ here?

I have the following equation (I'm actually deriving a deformation field from a strain tensor, but that's not important as at this stage it is pure calculus and I'm a little stuck at this stage): Ay ...
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### Table of Product Representations of Functions?

Does anyone know where I could find a (preferably free, online) table with infinite product expansions of basic functions (e.g. trig functions, logarithms, special functions)? Specifically ...
### If $f(x)=\sin(\log(\frac{\sqrt{4-x^2}}{1-x}))$ then the range of $f(x)$ is? [duplicate]
If $f(x)=\sin(\log(\frac{\sqrt{4-x^2}}{1-x}))$ then the range of $f(x)$ is? I found the domain of the function is $-2<x<1$.But I'm having difficulty in finding the range.