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

learn more… | top users | synonyms (1)

10
votes
1answer
58 views

Simple true/false statement about function composition

Given the functions $f,g$ from $\mathbb{R}$ to $\mathbb{R}$ is it true that If $f \circ g$ is strictly increasing and $f$ is injective then $g$ is monotonic I believe this is false but I can't ...
4
votes
3answers
74 views

What the terms “basis functions” and “orthogonal” denote in the case of signals?

I am a beginer. I have read that any given signal whether it is simple or complex one,can be represented as summation of orthogonal basis functions. Here, what the terms Orthogonal and Basis function ...
2
votes
0answers
49 views

Is there a way to follow the curve of a polynomial at a fixed speed?

I'm trying to follow the curve of a polynomial between $x=0$ and $x=1$ at a fixed speed using the arc length formula: $\int_0^1\sqrt{1+f'(x)^2}dx$ I've gotten around the square root problem by ...
0
votes
1answer
14 views

Is the relation $\Psi: M_2(\mathbb{Z^*}) \rightarrow \mathbb{Q}$ a Function

Determine if the following is a function Let $\Psi: M_2(\mathbb{Z^*}) \rightarrow \mathbb{Q}$ by $\Psi\big( \left[\begin{smallmatrix} a&0\\ 0&b\end{smallmatrix}\right]\big) = \frac{a}{b}$ ...
2
votes
1answer
41 views

Why is a “reverse surjection” incorrect?

In mathematics, it's perfectly fine if we have a function which maps multiple elements of a set $X$ to the same element of a set $Y$. Why would it be incorrect to map one element of $X$ to multiple ...
1
vote
1answer
16 views

Writing triangular function as a piecewise function

Given the following triangular function, I have to rewrite it as a piecewise function. $$x_\Lambda (t) = \sum_{n=-\infty}^{+\infty} \Lambda \left(\dfrac{t-nT}{T/2}\right), $$ where $T$ is a fixed ...
0
votes
1answer
18 views

Under What Conditions Is $f:M\rightarrow \mathbb{C}$ Where M Is the Set of 2x2 Matrices a Function and Not a Function?

I came across a problem that I thought was interesting. I attempted to solve the problem below, and I would be grateful if someone would check my logic in what follows. Let the set M of all 2 by 2 ...
1
vote
2answers
35 views

Is it true that $\frac{d}{dt}f(g(t),h(t))=f'(g(t),h(t))g'(t)+f'(g(t),h(t))h'(t)$

I want to solve the following question: We want to find $\frac{du}{dt}$ where $u(x,y)=x^2y^3$ and $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$. I know we can just plug $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$ in ...
0
votes
2answers
45 views

Show that $g$ is one-one if and only if $g$ is onto.

Original problem A function $g$ from a set $X$ to itself satisfies $g^m=g^n$ for positive $m$ and $n$ with $m>n$. Here $g^n$ stands for $g\circ g\circ \dots g$(n times). Show that $g$ is one-one ...
0
votes
1answer
14 views

Prove that left and right inverses are equal

Let $f:X\to Y$ be any function. Prove that if $h$ is a left inverse of $f$ and $k$ is a right inverse of $f$ then $h=k$ I was thinking of saying that: if $h$ is the left inverse that means ...
0
votes
3answers
49 views

Inverse function of $x^x$

How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried ...
0
votes
2answers
27 views

Confusion with the formal definition of injective function

The formal definition of an injective function is defined as $$\text{$\forall $a,b $\in $ A f(a)=f(b)$\Rightarrow $a=b}$$ My understanding of injective function is one that preserves ...
3
votes
1answer
16 views

Under what conditions is $ \max_{a \in A} (f(a) - g(a)) \geq \max_{a \in A} f(a) - \max_{a \in A} g(a) $ true?

Consider: $$ \max_{a \in A} (f(a) - g(a)) \geq \max_{a \in A} f(a) - \max_{a \in A} g(a). $$ Intuitively, it seems obvious it should be true, but I was having a hard time coming up with a rigorous ...
-1
votes
0answers
25 views

How to recall the domain, codomain and sketches of trigonometric functions.

Okay so I'm studying for a test and I'm always finding it a struggle to recall exactly what the domain and codomain of these functions are and recall their graphs. I know $\sin$, $\cos$, $\tan$, ...
1
vote
1answer
24 views

Showing topological properties of a function

Let $f$ be a function from a set $X$ into a set $Y$. prove: i) the function $f$ has an inverse if and only if $f$ is bijective ii) let $g_1$ and $g_2$ be functions from Y into X. If $g_1$ and $g_2$ ...
5
votes
5answers
133 views

Is it possible that $(f\circ g)(x)=x$ and $(g\circ f)(x)\ne x$?

Is it possible that $(f\circ g)(x)=x$ and $(g\circ f)(x)\ne x$ In other words, To show $f$ and $g$ are inverse, is it enough to show $(f\text{ o }g)(x)=x$? I have never witnessed a case in which the ...
7
votes
3answers
75 views

Pedantic question on function notation and the meaning of domain

Suppose we have a function $f: A\to B$. Then we know, without specifying what $f$ is, that $f$ may or may not map to every element $b\in B$. If $f$ does map to every element $b\in B$ then it's ...
1
vote
2answers
51 views

Seemingly easy Ordinary Differential Equation

For which values of $T$ can we find a unique solution of the ODE $x''(t) = −x(t) $ satisfying the boundary conditions $x(0) = a_1$ and $x(T) = a_2$ for any values of $a_1$ and $a_2$ ? I can solve ...
1
vote
1answer
40 views

Physical significance and graphical point of view of second derivative of a function $f''(x)$ .

This was just going through my mind - $f'(x)$ represents slope of a function. Then what does $f''(x)$ represent? For example, we define strictly increasing and strictly decreasing functions in some ...
0
votes
0answers
37 views

Basic Function Notation

I had a homework assignment due and I missed this question: $$f(5x) = f(x)^4 + 3 \times f(x) \times f(x)^2$$ with $$f(x) = 2x+1$$ I am really rusty but does this it mean this: $$f(5x) = ...
2
votes
3answers
42 views

For what real values of $a$ does the range of $f(x)$ contains the interval $[0,1]$?

Question : For what real values of $a$ does the range of $f(x) = \cfrac{x+1}{a+x^2} $ contains the interval $[0,1]$? My doubt lies in the further preceding of this question. The book states : ...
1
vote
3answers
43 views

Why is arcsin a function and not a relation since $\arcsin(\sin(\frac{3\pi}{4})) = \frac{\pi}{4}$?

Since $\sin(\frac{\pi}{4})$ and $\sin(\frac{3\pi}{4})$ are both $\frac{\sqrt{2}}{2}$, shouldn't $\arcsin(\frac{\sqrt{2}}{2})$ map to both $\frac{\pi}{4}$ and $\frac{3\pi}{4}$ and therefore not be a ...
0
votes
1answer
45 views

find the function f(r)

If $$W(r)= \frac{2r+1}{r(r+1)}$$ Express $W$ in the form $$W(r)= f(r) - f(r+1)$$ I tried doing this with partial factors but ended up in getting the + sign instead of - Please help
0
votes
1answer
24 views

Find a polynomial that satisfies the following conditions. [closed]

Let $f(x)$ is a polynomial of the degree $4$. Suppose $f$ verifies the following: $(x-1)$ is a factor of $f(x)$; When $f(x)$ is divided by $(x+1)^2$ the remainder is $-4(2x+1)$; When ...
0
votes
0answers
26 views

Can anyone help me find this function?

I am studying the concentration of a drug and need to create a model. The drug is taken at regular time intervals $T$. To do this I have decided to use a piece set function $h(x)$ for $nT < x < ...
2
votes
1answer
15 views

Solving all possible values for a functional composition

$f(x)$ and $g(x)$ are defined over the real number set $\Bbb R$ as follows: $$ g(x)=1 - x + x^2 \\ f(x) = ax + b $$ If $g ◦ f(x) = 16x^2 - 12x + 3$, determine all the possible values of $a$ and $b$. ...
0
votes
1answer
30 views

Complex function, analyticity domain

Find the function domain of analyticity i)$f(z)=\frac{z^2}{z-3}$ ii)$f(z)=ze^{-z}$ To check the domain of analyticity of a function, I only need to replace $z=x+iy$ and check the conditions of ...
3
votes
3answers
68 views

When does a function have an inverse?

I have been told that a function has an inverse if it is one-to-one or injective, but how can we rigorously prove this? I have been struggling to find a proof for days.
0
votes
0answers
29 views

Double integral problem 4

I can't solve both a and b. For part a I guess that the function can be 6x. Can anyone solve these problems?
8
votes
0answers
100 views

Exercise about continuous functions

Consider a continuous function $f \, : \, [0,1] \, \longrightarrow \, [0,+\infty)$ such that $f(0)=f(1)=0$ and : $\forall x \in (0,1), \; f(x) > 0$. I would like to prove that there exist ...
1
vote
1answer
58 views

Find the range of the given function $f$

Find the range of the following function. $$\cfrac{1}{\pi} \left(\sin^{-1} x + \tan^{-1} x\right) + \cfrac{x+1}{x^2 + 2x + 5} $$ where $\sin^{-1}x \ $ and $ \ \tan^{-1}x $ are inverse ...
0
votes
3answers
29 views

If graphs of the functions $y=ln x$ and $y=ax$ intersect at exactly 2 points then a must be?

If graphs of the functions $y=ln x$ and $y=ax$ intersect at exactly 2 points then a must be? Ok I know the graphs of these two functions but what should be the calculus based technique to find the ...
1
vote
3answers
14 views

Finding range of $||x| - |y||$ for the given conditions.

If $ z = x + iy$ and $ x^2 + y^2 = 16 $ , then the range of $||x|-|y||$ is...? This is what I've tried yet: Suppose $x = a\cos \theta$ and $y = b\sin \theta$, then we've : $$\begin{align} ...
1
vote
1answer
30 views

Find range of the given function : $ f(x) = \frac{e^x}{1+ [x] } $ when $ x \ge 0 $

Find the range for $ f(x) = \cfrac{e^x}{1+[x] } $ when $x\ge 0$ . Where $ [.] $ denotes greatest integer function. My book answers it in a very straight forward manner - Here f(x) is ...
0
votes
1answer
32 views

Suppose that a particle is located at the origin $(s = 0)$ at time $t = 1$ and moves along the line with velocity $v(t) = t^{-2}$.

Suppose that a particle is located at the origin $(s = 0)$ at time $t = 1$ and moves along the line with velocity $v(t) = t^{-2}$. How can I find the position s as a function of time? And how can I ...
0
votes
1answer
58 views

Proper notation for the function $g(x) = x^2+6$.

I'm using this more as a method of verifying if I'm correct on a question I am having difficulty with. Keep in mind, I'm a complete beginner, so.. yeah. Thereom: Assume the function $g$ is ...
0
votes
0answers
12 views

Function whose fixed points are a convergent sequence with derivatives at each term $\neq 1$ and derivative at limit point $=1$

The question asks to find an example of a function where the fixed points of the function are a sequence that converges to a fixed point, where the derivative of the function at each of the fixed ...
2
votes
0answers
33 views

Area & order-preserving function transformations

Consider a bounded function $f(x \in \mathbf{R}) \rightarrow \mathbf{R}$ with bounded support $\left[0,L\right]$ (illustration below). What type of transformations $g(f(x))$ guarantee that: Area ...
1
vote
2answers
57 views

Find whether $f(x) = x + \cos (x)$ is one-one.

Find whether $f(x) = x + \cos x$ is one-one. I tried finding $f'$ , $f'(x) = 1 - \sin x$ . But, how is this strictly increasing? Following is the graph for $1 - \sin x$ that I plotted using ...
2
votes
1answer
47 views

Where does this function come from in this proof?

This is an excerpt taken from a proof: Let each $M_n(n\in\mathbb{N})$ be countable, Then there exists an injective function $f_n:M_n\rightarrow\mathbb{N}$. Now, set a function ...
-4
votes
0answers
28 views

Vertical Distance Problem

Determine the two points on the $x$ axis at which the vertical distance between them and the respective corresponding points on the line $x-1$ is $6$ units. I understand how the difference function ...
2
votes
1answer
22 views

Find curve that fits (min, mean, max) to (0, 0.5, 1) [closed]

I'm trying to use the fact that $log(1) = 0$ and $log(\sqrt{e}) = 0.5$ and $log(e) = 1$ to write a map from a set of data points to a value between $0$ and $1$ such that: $f(min) = 0$, $f(mean) = ...
0
votes
0answers
29 views

Help with solving ODE differently [duplicate]

For which values of $T$ can we find a unique solution of the ODE $x''(t) = −x(t) $ satisfying the boundary conditions $x(0) = a_1$ and $x(T) = a_2$ for any values of $a_1$ and $a_2$ ? I can solve ...
1
vote
1answer
36 views

Whether the terms Bases function and Basis function are same or differen?

I have came accross few lines in my reading/discussion like any signal can be represented by summation of elementary Signals.these elementary signals are called as basis(sometimes read as bases?) ...
0
votes
2answers
32 views

Find an example of continuous but not increasing function whose inverse function doesn't satisfy the Inverse Function Theorem

I have to find an example of a function $f:[a,b]→R$ which is continuous, but not strictly increasing, such that no inverse function $f^{−1}$ satisfy the property of the Inverse Function Theorem.
1
vote
0answers
17 views

Parameterizing a linear compressor

I am hoping to build a function $f_{A,B,\alpha}(x \in \mathbf{R} ) \rightarrow y \in \mathbf{R}$ that serves as a positive signal compressor. The function acts on an input signal $x\left(t\right)$ one ...
2
votes
1answer
24 views

Is this function defined in terms of elliptic $\mathrm{K}$ integrals even?

Let $R,z > 0$ be positive real constants, and consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $$ f(v) = \frac{1}{\sqrt{(R+v)^2+z^2}}\ \mathrm{K}\!\left( \frac{4 R v}{(R+v)^2+z^2} ...
0
votes
0answers
33 views

Thinking Problem Involving Average Rate of Change

Here is the question: Suppose that the number of houses in a new subdivision after t months of development is modelled by $N(t)=\frac{100t^3}{100}+t^3$. where $N$ is the number of houses and $0\leq ...
1
vote
3answers
24 views

Is the function max{x,y} defined if x and y take equal values?

If x and y take the same values, will the function return a result? I am asking this as maximum means greatest of two values. So if both the values are equal, the existance of the function confuses ...
2
votes
3answers
238 views

Holomorphic function

Let $f(z)$ be a holomorphic function that maps the unit disk to the unit disk. Prove that $$|f^{(5)}(0)| \leq 120.$$ I use some concrete example it seems that this statement work out but i ...