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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2
votes
3answers
32 views

Inequality, only one solution from algebra

I recently came along the following problem: $$f(x) = {4-x^2 \over 4-\sqrt{x}}$$ Solve for:$$f(x) ≥ 1$$ My Attempt Now I know that one of the restrictions on the domain is $x≥0$, thus one of the ...
1
vote
1answer
14 views

Why does the average of a set of random numbers to the nth power approach 1/(n+1)?

I got bored and started running a Java program to mess with stuff like this. I did a boatload of trials and averaged them all together, first for a random number squared. Quick pseudo-code: ...
0
votes
1answer
14 views

Trouble while dividing a product

First of all sorry for my bad english. I stumbled upon this term. Is it possible to transform it like that? $$F\times r_1=\frac Q 2\times (r_1-r_2)$$ $$F=\frac Q 2\times \frac {(r_1-r_2)}{r_1}$$ ...
0
votes
1answer
41 views
1
vote
1answer
28 views

Solve $\log_9(x-4) - \log_9(x-8)= \frac{1}2$

Solve $\log_9(x-4) - \log_9(x-8)= \frac{1}2$ $(x-4) - (x-8)= 9^\frac{1}2$ $(x-4) - (x-8)= 3$ The answer is 10 but I am not sure how that was obtained.
1
vote
3answers
82 views

Maximum and minimum of of $f(x)=|x-1|+|x-2|+|x-3|$

I am trying to find the maximums or minimums of $$f(x)=|x-1|+|x-2|+|x-3|$$ (if there exist). My attempt: First I compute the derivative and tried to find critical point, i.e, $f'(x) = ...
2
votes
1answer
79 views

Solution of $x^2e^x = y$

The other day, I came across the problem (or something that reduced to the problem): Solve for $x$ in terms of $y$ and $e$: $$x^2e^x=y$$ I tried for a while to solve it with logarithms, roots, and ...
0
votes
0answers
46 views

Some basic manipulations with infinite series

In a linear algebra book that I am studying they prove that $$ ...
1
vote
2answers
27 views

Tridiagonal matrix inner product inequality

I want to show that there is a $c>0$ such that $$ \left<Lx,x\right>\ge c\|x\|^2, $$ for alle $x\in \ell(\mathbb{Z})$ where $$ L= \begin{pmatrix} \ddots & \ddots & & & \\ ...
-4
votes
0answers
58 views

$x\,|\, x$ is a positive integer number such that $x^{2} = 4$. [on hold]

Is the answer $2$? I have guessed that the value of $x$ is $2$ since $2^2 = 4$. Can it be so simple?
-1
votes
1answer
25 views

If $f(-f\circ f\circ\ldots\circ f(0)) - f(f\circ \ldots\circ f(0))\cdot f(-f\circ\ldots\circ f(0))$ what is $f(0)$

If $$f(-\underbrace{f(f(\ldots f(0)\ldots))}_{\text{n times}})-f(\underbrace{f(f(\ldots f(0)\ldots))}_{\text{n-1 times}})\cdot f(-\underbrace{f(f(\ldots f(0)\ldots))}_{\text{n-1 times}}) = f(0), ...
-1
votes
1answer
33 views

A monic polynomial which does not have a root in unit disc [on hold]

Let $P$ be a real polynomial of the form $P(x)=x^n+a_{n-1} x^{n-1}+\cdots+a_1 x-1$. Suppose that $P$ has no roots in the open unit disc and $P(-1)=0$. Then 1. P(1)=0 2. P(x) goes to infinity when x ...
7
votes
3answers
117 views

Evaluate $\int \frac{1-x}{(1+x)\sqrt{x+x^2+x^3}}dx$

Evaluate $$\int \frac{1-x}{(1+x)\sqrt{x+x^2+x^3}}dx$$ i used substitution $x=\tan^2 y$ so $dx=2\tan y \sec^2 y dy$ so the integral becomes $$I=\int\frac{2\cos 2y\: \tan y\: \sec^2 y ...
2
votes
1answer
39 views

Is there a solution to the absolute value of an expression which results in a negative value?

The equation given: $ \mid x - 4 \mid = -3$. My instinct (and example 2 in this article) tells me that there shouldn't be any solution as there would be no value of x which would result in a negative ...
0
votes
0answers
15 views

expression for a rational number

Let $x=\sum_{k=1}^n-b_kg^{-k}$ where $g\in \Bbb N$ and $b_k\in\{0,\cdots,g-1\}$. I want to write $x$ in the form $x=m+\sum_{k=1}^nc_kg^{-k}$ with $c_k\in\{0,\cdots, g-1\}$ and $m\in \Bbb Z$. Doing ...
1
vote
1answer
25 views

Need help understanding algebra steps taken in proof of why an even minus an odd is odd

I don't understand the algebra used in the below example proof from my textbook. Where does the + 1 come from? Is it okay to just add 1 anywhere you want? Or is there some rule here or reason you ...
0
votes
1answer
42 views
-1
votes
4answers
57 views

Solving for Roots/Zeros [on hold]

$$0=\frac{x}{4\sqrt{x^2+16}} - \frac{1}{10}$$ How would i go about solving this? I have started by bringing the -1/10 to the other side but after that I'm stuck and don't know what to do...
0
votes
0answers
35 views

If $f(-x-f(0))=f(x)+f^2(0)+2xf(0)$ is it possible to find $f(0)$

Given the functional equation $f(-x-f(0))=f(x)+f^2(0)+2xf(0)$ if we substitute $x=0$ we get $f(0)=f(-f(0))-f^2(0)$ but if we substitute $x=f(0)$ we get $f(-f(0))=f^2(0)+f(0)$, which is the same ...
0
votes
1answer
27 views

Solving a cubic function with P and Q

I have been struggling a little bit over solving cubic functions. I have been trying to use the P and Q method. So the question is What is the approximate value of the greatest zero of $f(x) = x^3 - ...
0
votes
2answers
30 views

Reducing algebraic fractions

$$\frac{ 9,009x^{4/3}y^2 - 7,007x^{7/3}y }{ 4,004x^{1/3}y }$$ $$\frac{ 1,001x^{4/3}y (9y - 7x^1) }{ 4(1,001)x^{1/3}y }$$ $$\frac{ x(9y - 7x) }{ 4 }$$ How is it ${ 7x }$ it looks like it should be ...
1
vote
2answers
35 views

How to find the quotient of a rational expression

I have been stuck on this problem because I don't know how to find the quotient of a rational expression. $$\frac{x^4 - 1}{x + 1}$$ Thanks
3
votes
2answers
50 views

If $f(-f(x))=f(-f(-x))$ can we conclude that $f(x)=f(-x)$?

If $f(-f(x))=f(-f(-x)), \quad f:\mathbb{R}\rightarrow\mathbb{R}$ can we conclude that $f(x)=f(-x)$? It seems unlikely, but I'am trying to solve a functional equation where the solution seems to be ...
1
vote
1answer
64 views

How do I show that $\frac {\cos^2 A}{\cos^2 B} + \frac {\cos^2 B}{\cos^2 C} + \frac {\cos^2 C}{\cos^2 A} \ge 4(\cos^2 A + \cos^2 B + \cos^2 C)$?

Let $A, B, C$ be the angles of an acute triangle. Show that $$\frac {\cos^2 A}{\cos^2 B} + \frac {\cos^2 B}{\cos^2 C} + \frac {\cos^2 C}{\cos^2 A} \ge 4(\cos^2 A + \cos^2 B + \cos^2 C).$$ How should ...
-6
votes
2answers
70 views

I want to solve this equation $e^{-x} = \ln x$ [on hold]

I wanted to solve this equation because I wanted to find intercept of the graph below.
2
votes
4answers
80 views

How do I show that $\sum_{i = 1}^n \frac 1{\sqrt{a_n}} \lt \frac {\sqrt 3}6$ for $a_n = 4n(4n + 1)(4n + 2)$?

Let $a_n = 4n(4n + 1)(4n + 2)$, show that $$\sum_{i = 1}^n \frac 1{\sqrt{a_i}} \lt \frac {\sqrt 3}6 \quad \forall n \in \mathbb{N}^+.$$ I know I need to find an upper bound for $1/\sqrt{a_n}$ but I ...
1
vote
1answer
41 views

How is the following proof really a proof (inequality)?

The user has just subtracted by $\frac{a}{b}$ in the first step and then rearranged the terms to show that it's positive and similar steps have been used to prove the second part of inequality. How ...
0
votes
2answers
76 views

What's wrong with my solution of inequality?

Question: solve the following inequality: $\frac{x}{2} \geq \frac{5}{x + 1} + 4$ My solution: $\frac{x}{2} - \frac{5}{x + 1} + 4 \geq 0$ $\implies \frac{x(x + 1) - 10(x + 1) - 8(x + 1)}{(x + 1) 2} ...
0
votes
1answer
27 views

When should one use a closed interval and when an open one in inequality?

In the following solution: In case I, the person has taken $2x \geq 0$ and then solved the equation. For the other inequality, he has taken $3 - x \gt 0$ and then solved the equation. My question ...
0
votes
2answers
23 views

Let $A\subset \mathbb{R}$ such that $l=\text{inf }(A)$ exists. Prove that $\forall \epsilon >0 $ there is $a\in A$ in the interval $[l,l+\epsilon)$

I need to prove the following: Let $A\subset \mathbb{R}$ such that $l=\text{inf}(A)$ exists. Prove that $\forall \epsilon >0 $ there is $a\in A$ in the interval $[l,l+\epsilon)$ That's what I ...
0
votes
3answers
69 views

Proving $\frac{n^2}{n-3}$ diverges

I need to prove that $$\frac{n^2}{n-3}$$ diverges For that, I need to prove that, given $\epsilon>0$, we have $n_0$ which depends on $\epsilon$, such that: $$n>n_0 \implies ...
3
votes
4answers
93 views

Real solution of the equation $\sqrt{a+\sqrt{a-x}} = x\;,$ If $a>0$

For a real number $a>0\;,$ How many real solution of the equation $\sqrt{a+\sqrt{a-x}} = x$ $\bf{My\; Try::}$ We can Write $\sqrt{a+\sqrt{a-x}} = x$ as $a+\sqrt{a-x}=x^2$ So we get ...
1
vote
3answers
41 views

Stirling on ${2n-1 \choose n}$

I'm trying to find an expression for $${2n-1 \choose n}$$ using Stirling's approximation $$k!\sim \sqrt{2\pi k}(\frac{k}{e})^k.$$ I see $${2n-1 \choose n}\approx ...
2
votes
0answers
112 views

Is $\dfrac{\cos\theta}{\sqrt{15}}$ irrational? [on hold]

In general I was wondering if $\cos\theta$ was between $0$ and $1$ exclusive then would $\dfrac{\cos\theta}{\sqrt{15}}$ be irrational? And just on another note is an irrational times a transcendental ...
0
votes
0answers
34 views

rational function cancellation [duplicate]

This is probably a trivial question however i cannot find the correct information online. When simplifying mappings from $\mathbb{R}$ to $\mathbb{R}$ such as: $$\frac{x(x-1)}{(x-1)}$$ Why is it ...
1
vote
1answer
59 views

how do I solve $y -\sin y= 1$

I am trying to use trigonometric equations to solve $y - \sin y = 1$, such as solving for $y$ but it is not working out, I have found $\cos y = \sqrt{-y^2 + y}$ but it does not lead to anywhere ...
1
vote
5answers
50 views

$x<y \iff x^{-1}>y^{-1}$

In order to prove the following: $$x<y \iff x^{-1}>y^{-1}$$ *for $x>0$ and $y>0$ I tried this: $$x<y\implies y-x>0$$ I have to prove that this, implies that ...
0
votes
1answer
38 views

Showing a function is odd

I have this equation: $$ f(x) = \frac{2x^2+3}{x-2} $$ and I have to prove it has half-turn symmetry around the point (2,8). I know that for a function to have half-turn symmetry, it needs to have ...
1
vote
3answers
31 views

Solve for a in Exponential equation

Is it possible to solve for $a$ in the following equation: $a^\alpha=b^\alpha-a$? Currently, I have resorted to using Excel to approximate $a$ (I am given values for $b$ and $\alpha$), but am ...
0
votes
1answer
32 views

Can someone show me the steps to simplify this?

$-x+x^{1.5}-2x^{1.5}+2x^{2}$ $=$ $2x^2-x^{1.5}-x$ I'm having trouble simplifying this. Can someone show me the steps?
0
votes
0answers
30 views

How useful is Prentice Hall Algebra 2?

I am currently a sophmore in highschool, and I wish to continue learning mathematics over the summer. My school laptop has a copy of Algebra 2 published by Prentice Hall. I have so far been unable to ...
1
vote
1answer
24 views

For what maximum positive $k$ is $2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$ true?

I am trying to find the maximum value of $k$ such that the inequality $$2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$$ is satisfied. I impose restrictions that $n \in \mathbb{Z}$ with $n \geq ...
1
vote
1answer
26 views

Finding parametric equations of rectangular equation

Is there a general process to follow when finding the parametric equations of a normal rectangular equation ? I know that one rectangular equation might have many parametric equations, but are there ...
-2
votes
0answers
19 views

differentiation- calculus [closed]

A farmer plans to construct an enclosure for his sheep making one side of a barn that is 150metres in length. Using 500 metres of fencing material the farmer will build a fence QRSTP which along with ...
0
votes
3answers
22 views

How can we find out the interval in an inequality?

Please go through the following link: Why is equating one of the bracks to zero in this equation correct? Now, the expression given there is $(x+1)(x+3)$, I understand now why we take either of these ...
0
votes
1answer
18 views

Explanation of homogenous function

Is there someone, who can explain why the function $g(s)=f(e^s,e^s)$ is not homogeneous when it can be written as $\frac{9}{4}e^{s/2}s$. I got the function $f(x,y)=\sqrt x +2\sqrt y +\frac{3y}{\sqrt ...
-2
votes
1answer
39 views

Find all possible solutions between 0 and 2π [closed]

I really could use help here, i am trying to find all possible solutions between 0 and 2 pi.
2
votes
1answer
133 views

A problem of olympiad. [closed]

This nice functional equation was proposed in the “VIII Olimpíada Iberoamericana de Matemáticas” held in Mexico (1993). Find all the functions $f:\mathbb N^* \to \mathbb N^*$ such that i) ...
2
votes
2answers
32 views

Squeeze Theorem with a restriction

Hi there I have this question: $$4x - 9 \leq f(x) \leq x^2 - 4x + 7$$ for $x \geq 0.$ Find the limit of $f(x)$ as $x$ approaches $4$ I know the answer is seven but why does the question inform me ...
1
vote
1answer
16 views

Construct sinusoidal functions

Can anyone explain how do I determine if the amplitude is positive or negative? I dont quite understand the explanation given here.