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

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1
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5answers
64 views

Proving that a function is an increasing function

Question: "5. Functions f and g, with domains $\mathbb{R}^{+}$, are defined as follows: $$\text{f}:x \to \sqrt{x}, \quad \text{g}:x \to 1 + 3x^{2}.$$ If the function h is defined by $h(x) = ...
0
votes
1answer
43 views

Solving a cubic equation

Solve $y=ax^3+bx^2+cx+d$ I need $x$ in terms of $y$ . I do not need the roots of the cubic equation . I need to express $x$ in terms of $y, x>0$
8
votes
5answers
935 views

What does orthogonality mean in function space?

The functions $x$ and $x^2 - {1\over2}$ are orthogonal with respect to their inner product on the interval [0, 1]. However, when you graph the two functions, they do not look orthogonal at all. So ...
2
votes
1answer
31 views

Smooth function from $\mathbb{R}^2\rightarrow\mathbb{R}$

I am asked if there exists a smooth function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ such that $f^{-1}(0)= \Delta$, where $\Delta$ represents the triangle with vertex $(0,0),(1,0),(0,1)$ I have ...
0
votes
1answer
30 views

Composition of functions involving the range

Consider the functions f, g A→ B→ C A → B = f, B → C = g Show that R(g ◦ f) = g(R(f)). (where R is the range). Now I think to show R(g o f) = g(R(f)) start with the definitions. x in R(g o f) if ...
2
votes
1answer
41 views

Derivatives of a function

I came across this problem and was wondering if I could get some guidance with this one? True / False. Every function f that is differentiable on the closed interval [a,b] is itself the derivative of ...
-1
votes
2answers
56 views

The inverse function of $x \mapsto 2x^3 + 5 $

I calculate the inverse of: $$ \Bbb{R} \to \Bbb{R}: x \mapsto 2x^3 + 5 $$ as: $$ \Bbb{R} \to \Bbb{R}: x \mapsto [(x-5)/2]^{1/3} $$ apparently it is not right, but I don't see where the problem ...
0
votes
0answers
10 views

Moving Logarithmic function equation plotted on log log paper up or down on the y axis

I'm hitting a stump here. I have a logarithmic function plotted on log log paper so it's a straight line. So let's say I have this entire line plotted out on the log log paper....how would I simply ...
0
votes
1answer
40 views

Does this special function exist?

Is it possible to find a non-trivial function $f(x_1,x_2)$ that has two parameters $x_1$ and $x_2$. This function should satisfy $f(x_1,x_2) = f(\frac{x_1}{1+r},x_2 +r)$, for any non-negative $r$. ...
2
votes
0answers
11 views

The $k$_th coefficient of the polynomial :$f_n(z)f_m(jz)+f_m(z)f_n(jz) $

Let $j=e^{2i\pi/3}$ ( $i$ is the complex number $i^2=-1$), and let : $$f_n(z)=(1+z)^n$$ Question Is there an expression (without using sums) of the $k$_th coefficient of the following polynomial ...
0
votes
0answers
14 views

How to write a periodic function expression from a piece of another function

How can I write a periodic function expression from a piece of another defined function?, e.g., how to write a periodic function using the piece of the function $f(x) = x^2$ in the interval $[-L,L]$. ...
1
vote
0answers
47 views

Help can't solve this one

Denote by $Q^{+}$ the set of all positive rational numbers.Determine all functions $ f : Q^{+}\to Q^{+}$ which satisfy the following equation for all $ x , y \in Q^{+}$ : $f(f(x)^2 y) = x^3 f(xy)$ I ...
0
votes
3answers
43 views

Finding an expression for $f^{-1}$ (function)

I need help with part (a) of this problem. Two functions, f and g are defined by $f(x) = \dfrac{x-1}{ x +1 }$ and $g(x) = mx+c$ (a) Find an expression for $f^{-1}$ I don't know how to make $x$ the ...
0
votes
3answers
26 views

Suppose that $f: X \rightarrow Y$ is a one to one function and $A \subseteq X$ then $Y - f(A) \subseteq f(X-A)$. Am I on the right track?

Suppose that $f: X \rightarrow Y$ is a one to one function and $A \subseteq X.$ $A, X, Y$, are all sets. I am trying to decipher if the following statements are true or false. If true I will need ...
0
votes
0answers
36 views

number of mod-3 monotone functions

I am looking for the number of mod-3 monotone functions of 1 variable. I am either using the wrong search criteria or there is no work on this particular area. A function f is called mod-3 monotone ...
-2
votes
2answers
43 views

Is $T(x,y)=(xy,0)$ a linear map? [closed]

Need to show that: $T = {R}^2 \rightarrow {R}^2 $ is not a linear transformation $T([ x, y]^ {T}) = [xy, 0]^ {T} = T $ Can you help get started.
0
votes
1answer
26 views

Given a collection of functions $f_i$ with the same domain, how to replace with values (w/o axiom replacement)

I know from a collection of ordered pairs we can project onto the first coordinate. I'm interested if there's a way (without using the axiom of replacement) to "project" a collection of functions onto ...
0
votes
1answer
23 views

Show that a function is a contraction in the metric d(x,y) = |lnx - lny|.

We have a function $f: \: (0,\infty) \rightarrow (0,\infty)$, and there is a constant $0<k<1$ s.t. $$x|f'(x)| \leq kf(x).$$ I want to show that $f$ is a contraction. Solving the differantial ...
1
vote
1answer
32 views

Is there a name for this relationship between two functions?

Is there a name for the relationship between $f()$ and $g()$ when $f(f(0,a),b)$ is guaranteed to be equal to $g(f(0,a),f(0,b))$ ? I'm using $0$ here to represent an initial state. The real problem ...
2
votes
1answer
155 views

Sum of resulting values of dice

We have thrown with $n$ dice. The sum of resulting values is $k$. We are looking for a function $f$ which gives the number of throws, with we can construct $k$ with $n$ dice. Some example for ...
0
votes
2answers
26 views

How to approach questions that ask to prove a function exists?

Consider the functions $r:S\rightarrow Q$ and $h:S\rightarrow T$ for arbitrary sets $S,T$ and $Q$. Prove that: if $$r(y)=r(x)\Rightarrow h(y)=h(x) $$ then we can find a function $g:Q\rightarrow T$ ...
1
vote
1answer
43 views

Mean Value Theorem problem

Given: $f:[0, 27] \to \mathbb R$ such that, $f(0)=0$ , $f(10)=1$ , $f(27)=1$ , where $f(x)$ is differentiable. Prove that , for some $\alpha$, $\beta$ $\in(0,3)$ , the relation $$2\int_0^{27} ...
0
votes
1answer
30 views

Is there a function $f: \mathbb{R} \to \mathbb{R}$ such that $f''$ is continuous and these properties $P(f)$ hold?

By $]a, b[$ we mean an open interval. Is there a function $f: \mathbb{R} \to \mathbb{R}$ such that 1) $f''$ is continuous; 2) $f'' > 0$ on $\mathbb{R}$; 3) $f'(0) = 1;$ 4) $f \leq 100$ on $]0, ...
5
votes
3answers
310 views

How to tell where parentheses go in functional notation?

The professor gave us a function $f(z) = \ln r + i \theta$ (this is for a complex analysis class). He doesn't like answering students' questions and there's no assigned textbook so I don't know where ...
0
votes
2answers
40 views

Is $g$ equal to $g'$: injective and surjective?

So the problem says that $f: X \to Y, g: Y \to Z$, and $g': Y \to Z$ are functions. Prove that $g\circ f = g'\circ f$ being that $f$ is surjective, then $g= g'$. So I understand that $f(x) = y,\ ...
1
vote
1answer
24 views

Finding an $f(x,y,n)$ such that $round[f(x,y,n)] = \lfloor\frac xn \rfloor + \lfloor\frac yn \rfloor$

Problem: I have an equation: $$\left\lfloor\frac xn\right\rfloor + \left\lfloor\frac yn\right\rfloor$$ I need to find an equation that does NOT use the floor function, but will take those same two ...
0
votes
1answer
20 views

Help: Question About Functions

Say we want to solve an equation like $2e^{f(0)}-(f(0))^2=2$ I would like someone to explain why the following procedure is wrong. I observe that $f(0)=0$ is a solution. If $f(0)=a$ is another ...
0
votes
3answers
36 views

Suppose that $f:X \rightarrow Y$ is surjective and $A \subseteq X$ then $f(X-A) \subseteq Y-f(A)$. True or False?

Suppose that $f:X \rightarrow Y$ is surjective and $A \subseteq X$ then $f(X-A) \subseteq Y-f(A)$. I am supposed to determine whether this statement is true or false. If true I am to prove it. If ...
0
votes
0answers
26 views

Find boolean function

Given $\mathbb{B} = \{true, false\}$, and function $f: \mathbb{B} \times \mathbb{B} \times \mathbb{B} \to \mathbb{B}, f(a,b,c) = a \land b \lor c,~ \forall a,b,c \in \mathbb{B}$. I want to find a ...
1
vote
1answer
21 views

Probability density function from the inverse of another function

Given the function: $$f(x) = 1/sin(x)$$ where x is the angular interval 0 ≤ x < 1.5708 (in radians). I want to obtain a probability density function which represents the inverse case of f(x). ...
0
votes
3answers
47 views

How do I come up with a continuous function between two functions?

Say $y = 0$ when $x \leq 0$, and $y = 1$ when $x \geq 1$. I want to create a function between these two that still makes everything continuous (continuous at $x = 0$ and $x = 1$) and is monotonically ...
2
votes
0answers
44 views

Find an equation in $x$ and $k$

Find an equation in $x$ and $k$ if, $$6u-8v+2=k^2$$ $$u^{2}=1+2v^{2}$$ $$v=2xy$$ $$u=x^2+2xy-y^2$$ Since we have 4 equations, we can eliminate 3 variables. But somehow, I'm not able to find an ...
1
vote
2answers
20 views

Find the Tangent Plane (Undefined?)

I've been asked to solve for a tangent plane at a point, but the method I'm using seems to lead to an answer that is undefined. Can anyone point me in the right direction with this? Write the ...
2
votes
1answer
43 views

Let $(X,d)$ be a metric space and $f:X\to X$ a function, is $d(x,f(x))$ a lower semicontinous function?

So I was trying to prove that if $f$ satisfies a special property the the function $d(x,f(x))$ is lower semicontinous but then I couldnt come up with a counter example of the following statement: Let ...
2
votes
3answers
33 views

what function fulfills these conditions? [duplicate]

So I know that if $f(x) = x^{-1}$, than $f(f(x)) = x$ but $f(x)$ is not necessarily $x$. So now, is there $g(x)$ such that $g(g(x)) \neq g(x) \neq x$ but $g(g(g(x))) = x$? If so what is it, else why ...
1
vote
2answers
50 views

Study of the first and second derivative of $\sqrt{|x^2+x|}-x$

I am not able to study the positivity of the first and second derivatives of $\sqrt{|x^2+x|}-x$ (that is, the values of $x$ for which the derivatives are positive, negative, or zero), because the ...
0
votes
3answers
32 views

Determine conditions for constants a,b,c,d so that $f \circ g$ = $g \circ f$

I have a homework problem that I don't know how to get started in: Let $f(x) = ax+b$ and $g(x) = cx+d$ Determine necessary and sufficient conditions on the constants a, b, c, and d so that $f \circ ...
3
votes
2answers
46 views

Homeomorphism from $(-1,1)$ to $\mathbb R$

I know that $f: (-1,1) \to \mathbb R$ defined by $f(x)=\tan \Big(\dfrac{\pi}2x \Big)$ is a homeomorphism . I am looking for some other homeomorphism between $(-1,1)$ and $\mathbb R$ which is not in ...
8
votes
3answers
1k views

Prove that f(x) = x

Let $ f : \mathbb{Q} \rightarrow \mathbb{Q} $ be a function which has the following property: $$ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x \;,\; \forall \; x, y \in \mathbb{Q} $$ Prove that $ f(x) = ...
1
vote
4answers
77 views

Is there a function $f$ such that $f'(x)=2x+f(x)$?

Is there a function $f:\Bbb R \to \Bbb R$ such that $f'(x)=2x+f(x)$? I've been trying to find it by inspection but I haven't found it, so now I'm wondering if it actually exists.
1
vote
3answers
49 views

For what values of $k$ is $g(x)=x^3+kx^2+x$ one-to-one?

I need to find for what values of $k$ $g(x)=x^3+kx^2+x$ is one-to-one. I tried finding for what values it is strictly increasing and got the derivative to be $3x^2+2kx+1>0$, but I'm not really sure ...
0
votes
1answer
29 views

Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions

Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and $f(x - y) = \sqrt{f(xy) + 1}$ for all $x > y > 0$. Determine ...
0
votes
1answer
54 views

Proving a function by induction [duplicate]

Let $f(n)$ be the function defined by $$ f(n) = \frac 1{\sqrt{5}} \left[ \left(1+\sqrt{5}\over2\right)^n- \left(1-\sqrt{5}\over2\right)^n \right] $$ How do you prove that $f(n) = f(n+2) - f(n+1)$ ...
0
votes
1answer
16 views

(product of) uniformly convergent functions and pointwise convergence

Consider 2 sequences of real functions on $I \subset \Bbb R$: $f_n \to f$ and $g_n \to g$ uniformly. Need to prove that $f_ng_n \to fg$ pointwise on $I$ From definition I know that $\forall ...
0
votes
1answer
28 views

Proving that a transformation of a function gives a positive result

If $x$ is real and: $$p = \frac{3(x^2+1)}{2x-1}$$ Prove that: $$ p^2-3(p+3)\geq 0$$ I think this has something to do with equating the discriminant to $0$, but I'm not entirely sure I'd really ...
0
votes
0answers
13 views

mobius transformation that sends $K = \{z\mid z= x+iy, x,y>0\}$ to $G$ which is an area bounded by two circles

Möbius transformation that sends $K = \{z\mid z= x+iy, x,y>0\}$ to $G$ which is an area bounded by two circles which intersect at $z = -1$ and $z = 1$. I tried sending $1$ to $1$, $i$ to $-1$ and ...
0
votes
0answers
29 views

Approximating continuous functions by steps functions: Proof that the approximation error monotonically decreases as the number of intervals increase

Let $f$ be a continuous function defined on a compact set, $f: X \subset \mathbb{R} \rightarrow \mathbb{R}$. Let $\mathcal{P}_k = P_1,\ldots,P_k $ be partitions of $X$ such that $\mathcal{P}_k$ is an ...
0
votes
0answers
23 views

Properties of functional calculus

Suppose we have a self-adjoint bounded operator $S$ on a Hilbert space $\mathscr{H}$ with the property that $||Sx||<||x||$ for each $x\in\mathscr{H}\setminus\{0\}$. Now assume that ...
0
votes
2answers
34 views

Modify tan(x) function to be sharper

On the right is -tan(x) + Pi/2 function. On the left is a function i am trying to create which is a "sharper" version of the function on the right. Any idea how ...
0
votes
0answers
27 views

Fastest way to check sign of different square

Given an image $I$ and two matrices $m_1;m_2$ (same size with I). The function $f$ is defined as: $$f=(I-m_1)^2-(I-m_2)^2$$ Because my goal design wants to get the sign of $f$. Hence, the function ...