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

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0answers
12 views

Minimum of a maximum function

What I'm trying to find is a linear function $y_{fit}$ ($y_{fit}=\beta_1+x\beta_2$) which minimizes the error $S= max(y - y_{fit}) - min(y - y_{fit})$. In other words, I'm looking for a linear ...
-4
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1answer
25 views

Determine the following function whether it is injective or surjective.

Determine the following function whether it is injective or surjective. $$f:\mathbb R^2\rightarrow\mathbb R$$ $$f(x,y)=x^2+y^2$$
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3answers
33 views

Suppose that $f: X \rightarrow Y$ is an onto function and $C$ and $D$ are subsets of $Y$ such that $f^{-1}(C)= f^{-1}(D)$. Then $ C = D$.

Suppose that $f: X \rightarrow Y$ is an onto function and $C$ and $D$ are subsets of $Y$ such that $f^{-1}(C)= f^{-1}(D)$. Then $ C = D$. I think this statement is false and I am trying to come up ...
-2
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0answers
48 views

Inverse in definition of a funtion

How to solve for the function f(x) ? Any help would be appreciated. $\ f(x)=1+\int_0^x(f^{-1}(a))da $ When I tried to solve for some number, like 5, in Mathematica, I wrote this: ...
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1answer
20 views

Determining whether a function is uniformly continuous

Determine whether $(4x-3)/(x-2)$ is uniformly continuous on the open interval $(1,2)$. I'm not sure how to start this as I have only answered these questions with closed intervals?
0
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1answer
25 views

Inverse of split functions

I don't know how to find the inverse is of a function when is split. Example, $\Bbb R_+$ is the set of positive real numbers. $f : \Bbb R \to \Bbb R_+$ $$f(x) = \begin{cases} 2-x & \text{if } ...
0
votes
2answers
33 views

Finding a bijective Map

I need to find a bijective map from $A=[0,1)$ to $B=(0,1).$ Is there a standard method for coming up with such a function, or does one just try different functions until one fits the requirements? ...
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0answers
10 views

Develop an equation and a set of 5 ordered pairs

For the following scenario, develop an equation and a set of 5 ordered pairs for equation: the amount a sales person earns is a flat rate plus commission. the volume of fuel in a snowmobiles tank ...
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1answer
32 views

Logic & Algebra - Trivia question [on hold]

If f(x) = g(x) * h(x - y) and g(x) = 2(x!) and h(x - y) = g(x) + h(y), what does f(4) equal when f(1) = -1, and y = x - 3?
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1answer
22 views

Suppose that $f: X \rightarrow Y$ is a function and $ B \subseteq Y$ is countable then $ f^{-1} $(B) is countable.

Suppose that $f: X \rightarrow Y$ is a function and $ B \subseteq Y$ is countable then $ f^{-1} $(B) is countable. My definition of countable is the following: A non-empty set $X$ is said to be ...
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2answers
21 views

The conditions that partial derivatives commute

State the conditions that partial derivatives commute, namely, $D_1D_2f = D_2D_1f$. I understand how to prove that these partial derivatives are equal but I don't understand what commute means. ...
0
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1answer
22 views

When will the population of the town be aprox $175000$?

By analyzing the impact of growing economic conditions, a demographer establishes that the predicted population, $P$, of a town $t$ years from now can be modeled by the following function: $$ P(t) = ...
1
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5answers
38 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
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1answer
30 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
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5answers
853 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
29 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
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1answer
20 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
36 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 ...
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2answers
52 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
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0answers
9 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
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1answer
39 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 $r$. ...
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0answers
9 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 ...
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0answers
11 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 $[-L,L]$. ...
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0answers
38 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
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1answer
17 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 ...
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3answers
20 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 ...
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0answers
21 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.
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2answers
41 views

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

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.
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1answer
25 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
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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
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1answer
31 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 ...
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1answer
65 views
+50

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 ...
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2answers
23 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
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1answer
42 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} ...
2
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1answer
27 views

To pick correct statements

I take $f = 1.5 x $ .This removes option B .But how do i check for other options .Only one is correct
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1answer
28 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, ...
0
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1answer
66 views

Simplify the expression $\left(1+ {1 \over 1+x}\right)\left(1+ {1\over 1-x}\right)$ [on hold]

I've decided to review back to functions, and for some reason, I'm confused!! I have to simplify this expression $$\left(1+ {1 \over 1+x}\right)\left(1+ {1\over 1-x}\right)$$ and I want to ask you ...
5
votes
3answers
302 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 ...
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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,\ ...
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1answer
23 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 ...
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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
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3answers
33 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 ...
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0answers
24 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
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1answer
19 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). ...
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3answers
46 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
40 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 ...
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2answers
18 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
39 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
30 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 ...
0
votes
0answers
20 views

Add lines between points in a Desmos graph [on hold]

I am currently making a desmos graphing calculator graph for a project. I have been able to create a table and rotate the figure represented by the table $a^\circ$ either clockwise or ...