Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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4
votes
3answers
143 views

Solve 3 exponential equations $z^x=x$, $z^y=y$, $y^y=x$ to get $x$, $y$, $z$.

The main question is : $z^x=x$, $z^y=y$, $y^y=x$ Find $z$, $y$, $x$. My method : I first attempted to get two equation for the unknowns $x$ and $y$. We can happily write : $z=x^{1/x}$ and $z=y^{...
0
votes
0answers
50 views

Simplify $F(x) = \exp[-\ln^2x^h]$

I was wondering if the expression $F(x) = \exp[-\ln^2x^h]$ can be simplified even further? As you can see, the $\ln$ which is the natural logarithmic function is raised (and not its argument) to power ...
-4
votes
0answers
30 views

Solve for the conditions given below [closed]

\begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions $f$, $g$, $h$ are defined for all $a,b,c,d\in\mathbb R$. For instance: $h$ can be Division; $a$, $b$, ...
2
votes
1answer
75 views

What is the value $f(-4)$ in the under function such that $f(x)+f(\frac1x)=\frac{x^2-12x+1}{2x}.$

Let $f$ is a function such that $$f(x)+f(\frac{1}{x})=\dfrac{x^2-12x+1}{2x}.$$ Then what is the value $f(-4)=$?
0
votes
2answers
38 views

Does this set of coordinates result in a curve?

Coordinates: (0,0), (3,3), (6,4.5), (9, 5.25) If this is a curve is there a formula for determining the y value for any given x within the range 0 to 9?
2
votes
2answers
41 views

Domain of $f(x)=x^{\frac{1}{\log x}}$

What is the domain of $$f(x)=x^{\frac{1}{\log x}}$$ Since there is logarithm , the domain is $(0 \: \infty)$ But the book answer is $(0 \: \infty)-\{1\}$ but if $x=1$ $$f(x)=1^\infty=1$$ So is it ...
1
vote
2answers
24 views

Why does this equality work when k, N, and r are all positive?

The expression is $\frac{r^N - \left( r-\epsilon \right )^N}{r^N}=1 - \left ( 1- \frac{\epsilon}{r} \right )^N$. I understand where the first $1$ comes from, but where does the $\left ( 1- \frac{\...
4
votes
2answers
201 views

How to approach general solutions to functional equations of multiple variables

I understand the concept of a function, broadly speaking, but when it comes down to solving general functional equations, I sometimes find it difficult to wrap my head around the problem at hand. For ...
5
votes
1answer
52 views

Bound on $c-b$ for $a^n+b^n=c^n$

Let $a\leq b\leq c$ be positive real numbers and $n$ positive integer with $a^n+b^n=c^n$. Prove that $c-b\leq(\sqrt[n]{2}-1)a$. The desired inequality can be written as $c-b+a\leq \sqrt[n]{2}a$. ...
1
vote
2answers
39 views

Is it correct this reasoning?

Let $E,F$ be reals vector space. Since (1) $\dim (E\times F)=\dim E + \dim F$ (2) $\dim\ \text{Hom}(E,F)=\dim E\cdot \dim F$ Given $r>0$ integer, is it true that: $$\text{Hom}(E\times \stackrel{(...
1
vote
5answers
94 views

How to decompose $x^3-1$

I need to decompose $x^3-1$, I know the Binomial theorem, and finding roots of a polynomial, how should I approach this?
1
vote
1answer
65 views

What is the equation of this graph?

This will sound very dumb, but I want $1000$ coordinates of this shape: How can I do that?
3
votes
2answers
65 views

Represent $\dfrac{\lambda_1^M-\lambda_2^M}{\lambda_1-\lambda_2}$ in terms of $\lambda_1+\lambda_2$ and $\lambda_1\lambda_2$

I have a problem as follows: Let $\lambda_1, \lambda_2$ are roots of the equation $\lambda^2-a\lambda+b=0.$ It can be proved easily (by induction for example) that the quantity $$\dfrac{\lambda_1^M-\...
0
votes
1answer
15 views

Algebra and summation question

${(1+\frac{q_jr^i}{1-q_j})}^{-1}=\sum_{k=1}^{\infty} (-1)^{k-1}(\frac{q_jr^i}{1-q_j})^{k-1}$ What rule is being used to go from the LHS to the RHS? My knowledge in maths is first year undergrad, but ...
-1
votes
0answers
23 views

Wanting to reverse an equation to determine d

I am wanting to please get some assistance to reverse this calculation so that I can determine d based on a varying Q. Apologies for the shockingly written and presented equation.. $$Q = \frac {4....
0
votes
1answer
49 views

When are we permitted to multiply or divide both sides of an equation by a constant?

For example, let's consider the quadratic equation $-3x^2 + 6x -2 = 0$. Multiplying both sides by $-1$, we get the equation $3x^2 - 6x +2 = 0$. The graph of the above equations are different even ...
-2
votes
0answers
20 views

Functions Explanation [closed]

The volume, $V$, in liters, of water coming out of the hose in $m$ minutes is given by the function $V(m) = 20m$ What does the constant 20 represent in the question (Rate//slope?? of water) Thanks.
-3
votes
1answer
45 views

Trouble simplifying the following expression. [on hold]

Let $x = t \cos(2t)$ and let $y = t \sin(2t)$. Now show the following equation is true. $$-200xe^{-x^2-y^2} (\cos2t - 2t \sin2t) - 200ye^{-x^2-y^2} (\sin2t +2t \cos2t) = -200te^{-t^2}$$ ...
-3
votes
0answers
25 views

Need equations based on the following rules. [closed]

I have a problem. I'm making a scoring system based on multiple input parameters that's probably over-complicated but it works like this: Low score if most parameters are low. High score if most ...
0
votes
1answer
23 views

$2k-1$ is an odd integer if $k$ is an integer

I am working on this advanced power rule problem: This is the image of the problem I understand everything up until step 4 in the problem hint. I am getting stuck with the statement that says: "...
5
votes
0answers
72 views

How to find area of a polygon built on the roots of a given polynomial?

How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane? For example, consider the equation: $$2x^5+3x^3-x+1=0$$ It has one real and ...
0
votes
1answer
29 views

Find parameter m knowing that the values of the function are in a interval of length 4

Please give me a hint on how to find the parameter $m$ knowing that the function values are in an interval of length $4$: $f(x)=\frac{x^2 + mx + 1 }{x^2-x+1}$.
0
votes
5answers
48 views

Inverse Equation of the Given Equation

Having a bit of a problem getting the inverse of the following equation: $$f(x) = \sqrt{9-x^2}$$ I had an answer which was equal to $3-x$ but when I used sites like Mathway and Wolfram to check my ...
1
vote
1answer
74 views

Find the values of $b$ for which the equation $2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$ has only one solution

Find the values of 'b' for which the equation $$2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$$ has only one solution. =$$-2/2\log_{5}(bx+28)=-\log_5(12-4x-x^2)$$ My try: After removing the ...
1
vote
2answers
51 views

Find the value of $P(1)$

Let $P (x) = x^2 + bx + c$, where $b$ and $c$ are integer. If $P(x)$ is a factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^2 + 28x + 5$, find the value of $P(1)$. I am not being able to solve ...
1
vote
1answer
34 views

Quadratic Equation Based Problem:Prove either $a = 2l$ & $b = m$ or $b + m = al$

If by eleminating $x$ between the equation $x² + ax + b = 0$ & $xy + l (x + y) + m = 0$, a quadratic in $y$ is formed whose roots are the same as those of the original quadratic in $x$. Then ...
0
votes
1answer
27 views

What's the relation between earth coordinates and angles?

I've been looking for an answer for a specific question, a part of my question maybe related to this: Calculate the angle of a vector in compass (360) direction However, my question is more specific, ...
1
vote
0answers
42 views

Proving $a^ma^n=a^{m+n}$ by induction when $n$ or $m$ is negative (or both)

Suppose we have already proved this exponent law for when $m,n\in\mathbb{Z^+}$ as in here. Also suppose $x^{-n}=\frac{1}{x^n}$ is given as a definition. Let $m=-\lambda$ and $n=-\gamma$, where $\...
0
votes
2answers
50 views

Finding the sum of $\cos45°$ + $i\cos135°$ + … + $i^{n}\cos(45+90n)°$ + … + $i^{40}\cos3645°$

My question is as follows: If $i^{2}$ = -1, find the value of $$\cos45° + i\cos135° + \ ...\ + i^{n}\cos(45+90n)° + \ ...\ + i^{40}\cos3645°$$ without the aid of a calculator. In terms of my attempts ...
1
vote
2answers
48 views

Prove that this is one-one, but not onto $\Bbb R$.

$\Bbb R$ stands for real numbers. $ f(x) = \begin{cases} 2-x, & \text{if $x \le 1 \qquad \text{is one to one but not onto } \Bbb R $ } \\ \frac{1}{x} , & \text{if $x >1$ } \end{cases}...
2
votes
3answers
128 views

Evaluate $\cos 36^\circ - \cos 72^\circ$ without the aid of a calculator [duplicate]

I have a quick question about a difficult trigonometric functions problem that I have been assigned. The problem is as follows: Evaluate $$\cos36° - \cos72°$$ without the aid of a calculator. In terms ...
2
votes
1answer
66 views

Inequality on a sequence of $n$ reals whose sum is $0$

Consider $n\geq3$ real numbers $a_1,a_2,\dots ,a_n$ satisfying $a_1+a_2+\cdots+a_n=0$ and $$2a_k \leq a_{k-1}+a_{k+1}$$ for all $2\leq k\leq n-1$. Prove that $$|a_k|\leq\frac{n+1}{n-1}\,\max\big\{|a_{...
0
votes
0answers
30 views

calculating moments in a table

I am trying to calculate the moments in a data list position data 1 15 2 22 3 5 4 2 5 1 to find out where in the list is ...
0
votes
2answers
56 views

Why can't z = 0 in this rational expression?

I came across this expression, which I was asked to simplify and then choose the number that would make the expression undefined: $$\frac{17z^3+17z^2}{34z^3-51z^2}$$ I simplified the expression to $\...
3
votes
8answers
90 views

How to prove the inequalities between $20^{70^2},30^{60^2},40^{50^2}$

Let $$M=\{ 20^{70^2}, 30^{60^2},40^{50^2}\}$$. What number is the greatest and which is the smallest? I thought about beginning by assuming certain inequalities and trying to prove them, for example: ...
0
votes
1answer
34 views

Sides of triangle are in A.P., find its perimeter

The sides of a triangle are in Arithmetic Progression $(A.P.).$ If the smallest angle of the triangle is $\alpha$ and largest angle of the triangle exceeds smallest angle by $\beta$ , then what is the ...
2
votes
2answers
50 views

Find a point on $y=\frac{1}{x^2}$ such that $y'=16$

I'm very new in this forum and I hope I don't ask something silly, which is asked many times before. I have to answer this question: Find the coordinates of the point(s) at which the curve has ...
2
votes
2answers
26 views

How to Solve $2.3856 + \log r = \log(364r - 363)$

I am solving geometric sequence and series problem, but got stuck on the logarithm part. we haven't tackled logarithm yet so this is supposed to be a challenge problem. can anyone advise on how to ...
1
vote
3answers
44 views

Progressions modulo $n$

I don't understand how to do these 2 tasks: 1) Prove that any arithmetic progression modulo $n$ has a period that divides $n$. 2) Prove that any geometric progression modulo a prime number $p$ has a ...
0
votes
0answers
25 views

Percentage of votes received in an election

In an election, $70\%$ of males were registered voters and $40\%$ females were registered voters, all registered males casted their votes. But only $65\%$ registered females casted their votes. If ...
1
vote
2answers
33 views

How to solve this inequality problem?

Given that $a^2 + b^2 = 1$, $c^2 + d^2 = 1$, $p^2 + q^2 = 1$, where $a$, $b$, $c$, $d$, $p$, $q$ are all real numbers, prove that $ab + cd + pq\le \frac{3}{2}$.
-1
votes
1answer
69 views

Is there a basis which spans the real numbers?

Is there a finite set of real numbers $S=\{a_1, a_2, ..., a_n \}$ such that every real number can be written as a linear combination (with integer coefficients) of the elements of $S$? If no, is there ...
2
votes
1answer
58 views

Sum of series $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$

I need to find sum of the series involving cube roots of unity $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$. Found it in an old test paper. I applied Geometric Progression Sum Formula....
1
vote
1answer
31 views

The greatest common divisor of $(O_n, T_n+2)$ where $O_n$ and $T_n$ are the oblong and triangular numbers respectively.

Suppose that $T_n$ is odd. Can we find infinitely many $n$ such that $(O_n, T_n+2)=1$? Is it trivial and obvious? My hunch based on some hand calculations is to look at $n$ congruent to $0$ or $2$ ...
1
vote
3answers
49 views

Find the following one-sided limits?

$$\text{a)} \ \ \lim_{x\to-2^+}(x+3)\frac{|x+2|}{x+2}$$ $$\text{b)} \ \ \lim_{x\to-2^-}(x+3)\frac{|x+2|}{x+2}$$ The answers are: $$\text{a)} \ \ 1$$ $$\text{b)} -1$$ How do you find them? It is ...
5
votes
1answer
35 views

Given a finite sequence, can we always find a relation that generates that sequence?

This is just something I've been wondering about, but I have no idea what the answer is. I suspect it's yes. Given an arbitrary finite sequence, can we always find a relation that generates that ...
2
votes
1answer
70 views

Why must $|z|\gt 1$ be the necessary condition

Question:- If $\left|z+\dfrac{1}{z} \right|=a$ where $z$ is a complex number and $a\gt 0$, find the greatest value of $|z|$. My solution:- From triangle inequality we have $$|z|-\left|\dfrac{1}{...
0
votes
2answers
125 views

How does $x^4+y^4=z^2 \implies x^4+y^4=z^4$?

Why is the statement "the following cannot be satisfied" for $x^4+y^4=z^2$ more strong than for $x^4+y^4=z^4?$ More specifically, how does $x^4+y^4=z^2 \implies x^4+y^4=z^4?$ This statement was ...
2
votes
0answers
37 views

Minimize a huge two-variable logarithmic-trigonometric-radical expression (MSU entrance early July 2016)

Minimize \begin{align}R(a,x)&=\sqrt{13+\log_a\left(\cos\left(\frac xa\right)\right)^2+\log_a\left(\cos\left(\frac xa\right)^4\right)}+\sqrt{97+\log_a\left(\sin\left(\frac xa\right)\right)^2-\...
4
votes
1answer
95 views

An inequality involving two complex numbers

Let $z_1, z_2 \in \mathbb C$ and $a,b \in \mathbb{R} \setminus \{0\}$. Prove that $$|z_1|^2+|z_2|^2-|z_1^2+z_2^2|\le 2\dfrac{|az_1+bz_2|^2}{a^2+b^2}\le |z_1|^2+|z_2|^2+|z_1^2+z_2^2|$$ ...