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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2answers
43 views

Prove the given condition from given two quadratic equation

Question: If the quadratic equations $x^2+bx+c=0$ and $bx^2+cx+1=0$ have a common root then prove that either $b + c + 1 = 0$ or $b^2 + c^2 + 1 =bc + b + c$ Till yet, I had figured the common ...
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2answers
73 views

Difficult algebraic expression for $f(x) = \frac{x-a}{bx-c}$ to find involutory solution

So I read in another thread about involutory functions, he claims for any real numbers $a$ and $b$, the function: $$f(x) = a + \frac{b}{x-a} = \frac{ax + (b-a^2)}{x-a}$$ satisfies $$f(f(x)) = a + \...
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4answers
131 views

Spivak Sine confusion (possible error)

quote from Spivak: "Let us consider the function $f(x) = \sin(1/x)$." The goal is to show it is false that as $x \to 0$ that $f(x)\to 0$ He says we have to show "we simple have to find one $a > 0$...
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1answer
48 views

Simplification of a formula with several exponential functions

I'm trying to work this problem my teacher gave as practice, I have the answers but I'm not sure what I'm doing wrong. I'd be grateful if anyone could help me out. Thank you so much.
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3answers
78 views

How can we prove that the square root of any number is equal to the statment given below

Is there any theorem that can explain this $\sqrt[n]{x}= x^{1/n} $, is there any practical example of $ x^{1/n} $, 1/n times of a number.
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2answers
34 views

Evaluating product of exponent and polynomial

In a probability theory problem, I need to solve an inequality over $n\in\mathbb{N}$ which can be expressed in a general form like this: $a^n ( b_1 n^2 + b_2 n + b_3) \leq c$ where $a, b_1, b_2, b_3,...
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1answer
150 views

Possible length of isosceles triangle side.

The perimeter of a right triangle $RST$ is equal to the perimeter of isoceles triangle $xyz$ The lengths of the legs of the right triangle are 6 and 8. If the length of each side of the isoceles ...
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1answer
44 views

Which has the least value?

Which of the following have the least value if $-1 < x < 0$? (A) $-x$ (B) $1/x$ (C)$-1/x$ (D)$1/x^2 $ (E)$1/x^3$ I'm not sure what to do, but I'll definitely try. We can break ...
2
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2answers
193 views

Solving $ \frac{1}{ a} = \ \frac{1}{ \ \sqrt{b}} \ +\ \frac{1}{ \ \sqrt{c}} $ with additional conditions

How to solve this equation $$ \frac{1}{a}= \ \frac{1}{ \ \sqrt{b}} \ +\ \frac{1}{ \ \sqrt{c}} $$ where $$ b = \sqrt{ (x-a/2)^2 + y^2 + z^2 )}$$ & $$ c = \sqrt{ (x+a/2)^2 + y^2 + z^2 )}$$ ...
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1answer
26 views

Checking some work on finding roots

OK, I have the following response function: $$H(\omega) = \frac{1-\omega^2 LC}{1+\omega^2 LC - i \omega RC}$$ I want to find where it becomes $\frac{1}{\sqrt{2}}$. This should be simple enough. ...
0
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1answer
55 views

Logarithms in Calculators?

I have no idea how to do logarithms, or even what they are, but our class recently received an extra credit problem pertaining to one. This helps me with EXACTLY what I want to do, but I have no idea ...
0
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3answers
80 views

Injectivity in function $f(x)=x\cdot|x|+1$

I want to prove that $f(x) = x\cdot|x|+1$ is injective, and if it is; find the inverse of the function. $f(a) = f(b) \iff a|a|+1 = b|b|+1 \iff a|a| = b|b|$ $\begin{cases} -a^2 = b^2 \quad undefined \...
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2answers
44 views

What is important to know in regards to trig functions?

I believe I forgot everything I learned in pre calculus 3 years ago, and I need to fine tune my studies. I just took a look at the book I will be using this spring and it has a few questions stating $\...
2
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4answers
189 views

Solve the equation $4^{7 x - 10} = 10^{7 x - 6}$ for $x$

Solve for $x$ of the following equation. $$4^{7 x - 10} = 10^{7 x - 6}$$ I tried to make $4$ and $10$ have a common base but I could not find one so I don't know where to go from here.
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1answer
110 views

find two equations for the tangent lines to the curve

So I have one answer but I can't figure out how to get the other answer so I was seeing if someone could help me out here. Find equations of the tangent lines to the curve $\displaystyle y=\frac{x-1}{...
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1answer
31 views

How to solve this system of 3 equations with substitution?

I have the system: $$-4 + λ = -3a + at\\ 1+2λ = -a + at\\ 3λ = 3a-at$$ but whenever I try to substitute, I end up getting lots of fractions that are hard to work with. By summing the 2nd and 3rd ...
0
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1answer
137 views

Interpreting 3 Circled Venn Diagrams

The Venn diagram above represents the 20 students who took one or more of the 3 available art classes or took no art classes at all. two students took no classes at all. 1.How many people take only ...
0
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1answer
37 views

Simplification of another nasty expression

I have the following condition $$ 2 \frac{x^2}{y^2} \left(1 - \frac{1}{y^2} \right)+ \frac{1}{y^2} \leq 1$$ Can anyone help me simplify it to the best possible relationship between $x$ and $y$?
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1answer
46 views

Unclear step in a textbook trigonometric identity proof

This is a step in the proof of a trigonometric identity: $$\frac {1+cos\left(\frac {\pi}{2}-a\right)}{1-cos\left(\frac {\pi}{2}-a\right)}=\frac {2\cos^2\left(\frac{\pi}{4}-\frac a2\right)}{2\sin^2\...
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1answer
54 views

how many questions did D answer correctly

Each of A, B, C, and D took a test. Each of them answered at least one question correctly, and altogether they answered 67 questions correctly. A had more correct answers than anyone else. B and C ...
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4answers
174 views

Find the exact value of $\sin\left(\arcsin(0.5)+\arctan(-4)\right)$

Find the exact value of $\sin\left(\arcsin(0.5)+\arctan(-4)\right)$ My calculator gives a decimal for $\arctan(-4)$ so I don't know what answer is expected.
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1answer
603 views

Bearings and distances

Two ships leave port at the same time. One travels at $5$ km/h on a bearing of $46$ degrees. The other travels at $9$ km/h on a bearing of $127$ degrees. How far apart are the ships after $2$ hours?
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1answer
70 views

Confused by the solution of $x^3+bx^2+cx+d=0$

From $x^3 + bx^2 + cx + d = 0$, we have $(x-x_1)(x-x_2)(x-x_3)=0$ for some roots $x_1$, $x_2$ and $x_3$. Expanding this second expression gives us $$x^3 + \left(x_1+x_2+x_3\right)x^2 + \left(x_1x_2 + ...
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2answers
39 views

Finding the value of an expression with logarithms

Given that $\log_{b}a=0.74$ and $\log_{b}(a-1)=0.65$ find the value of the following expression: $$\log_{b}(a^{4}-1)-2\log_{b}(a^{2}+1)+\log_{b}(a^{3}+a)-\log_{b}(a+1)$$ I tried using log laws to ...
0
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2answers
30 views

Simplify an expression involving indices

One of my friends asked me this question: Simplify $$\frac{50^{3x-1} 10^{2-3x}}{250^{3x+1}}$$ I've been thinking about the question for more than a day. I've looked through my teacher's notes but ...
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3answers
4k views

Can a sum of products be split as a product of two sums?

I have $$\sum_k^n P_k x_k$$ Am I allowed to split it up into two sums so I have it like $$\sum_k^n P_k \sum_k^nx_k$$
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3answers
27 views

The relation between hyperbolic sine and hyperbolic cotangent

I was wondering if someone can verify (or not) the correctness of the following function? $$\frac{1}{\sinh^2X}=\coth^2X-1$$ I saw it in a paper but I am weak in math, so I am unsure if it is correct ...
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1answer
114 views

If $x^2+y^2+z^2=3/2$, then $\sum\frac{x^2}{x(4x-3)+z^2+y^2} \le1+\frac{\sqrt2}2$

If $x$, $y$, and $z$ are real numbers satisfying $$ x^2 +y^2 +z^2 = 3/2 $$ then prove that $1+\frac{\sqrt{2}}{2} \geq$ the cyclic sum of $$ \frac{x^2}{x(4x-3)+z^2+y^2} . $$ I've tried Cauchy-...
0
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2answers
88 views

Prove that if $p$ is a prime and $k$ is an integer, there are two integers $x$ and $y$ that satisfy $x^{2} + y^{2} + k \equiv p$ [closed]

Prove that if $p$ is a prime and $k$ is an integer, there are two integers $x$ and $y$ that satisfy $$ x^2 + y^2 + k \equiv 0 \pmod p. $$
1
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2answers
82 views

Solve $\cos(5y) + \cos(3y) +\cos(y) = 0.5$ for real $y$.

Well $\cos(3y)=\cos(y+2y)=\cos(y)\cos(2y)-\sin(y)\sin(2y)$. That's all I got. I've tried putting it in the equation but it doesn't seem to work out. How to solve this?
0
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1answer
54 views

What is the order of operations when solving for $\ f\circ f \ $ if $f(x) = x + \frac{1}{x} $

I am little confused as to how I can solve rational polynomials such as $\ f\circ f \ $ if $f (x) = x + \frac{1}{x} $. $$f(f(x)) = x + \frac{1}{x}+ \frac {1}{x+1/x}$$ Am I only allowed to ...
0
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2answers
201 views

Is there a general method for solving a cubic polynomial? [duplicate]

I'm doing a course in linear algebra at the moment, and whenever I need to find the eigenvalues of a 3x3 matrix, I'm faced with the issue that I don't know a general method for solving a cubic ...
1
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2answers
41 views

Find the function and domain for $ (f\circ f)$ when $ f(x) = x+ \frac {1} {x} \ $

Find the function and domain for $ (f\circ f)$ My answer is $ \frac {x^4+3x^2+1x} {(x^2+1)(x)}?$ However, the program I am using states I am wrong. What have I done incorrectly?
2
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3answers
208 views

Unclear step in the proof of half-angle formula for tangent

I wonder how could $$2\cos^2\left(\frac a2\right)$$ be transformed into $$1+\cos(a)$$ This is from a step in my textbook's proof of the tangent half-angle formula: $$tan\left(\frac a2\right) = .. =\...
3
votes
4answers
675 views

Proving $ \binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n $ without induction [duplicate]

I have to prove that: $$ \binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n $$ I don't want a complete solution, but only a hint.
0
votes
1answer
77 views

Find all real x such that $\cos x$, $\cos2x$, $\cos 4x$, $\cos 8x$, etc. ($\cos 2^n$ for all non negative $n$) are all negative

I think I got that $|\cos (2^nx)|$ must be less than $|\sin (2^nx)|$ for all non negative $n$.
8
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1answer
230 views

How to prove that $\frac{1}{x_1}+\frac{1}{x_2}+…+\frac{1}{x_n}-\frac{1}{x_1x_2…x_n}\in \mathbb{N}\cup \{0\}$

Question: Show that for every natural number $n$ there exist $n$ natural numbers $ x_1 < x_2 < ... < x_n ,$ such that $$ \frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}-\frac{1}{x_1x_2......
10
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2answers
259 views

Solving the functional equation $f(xy)=f(f(x)+f(y))$

Find all functions from $f: \mathbb{R} \to \mathbb{R}$ such that for all $x$ and $y$ $$f (xy)=f (f (x)+f (y))$$ I've put $x$ and $y$ as $0$ and $1$. How to proceed after substituting if we don't ...
0
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1answer
76 views

Simplification of nasty expression

I have the following equation which I am trying to solve, $$ \frac{x^2}{y^2}- \frac{x^2}{y^4} -\frac{1}{2} \leq 0$$ Can anyone think of a way of simplifying the above, I don't think this is a form ...
1
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1answer
62 views

Prove inequality formula by induction

my question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 35. Exercise 1. Prove the following formula by induction: $$1^3+2^3+3^3+\cdots+(n-1)^3<...
0
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3answers
580 views

What is the meaning of $\log^2n$ and how should it be read in word form?

$\log^2n$ is what I need assistance with. How is this read in word form? What exactly does this mean? No matter how much I read about logarithms, they still seem new to me.
-1
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1answer
194 views

If the quadratic equation $x^2 + 2kx + 2(k + 4) = 0$ has distinct real roots, then $k^2 – 2k – 8 > 0$ [closed]

The quadratic equation $x^2 + 2kx + 2(k + 4) = 0$ has distinct real roots. Show that $k^2 – 2k – 8 > 0$. I'm not sure what you're meant to do here- it's a 2 mark question.
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2answers
74 views

Solving equations with fractional indices

How would I go about solving an equation like this? $3^{4/3}b^{5/3} - b^3 = 1$ I thought about rearranging to get $3^{4/5}b = (1 + b^3)^{3/5}$, but that didn't seem to lead anywhere as I couldn't ...
1
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2answers
73 views

Factor $x^6+\dots+1$ over $\mathbb R$

Out of idle curiosity, while teaching Calculus II, I started to wonder about this: How do you factor the polynomial $$ f_6(x)=x^6+x^5+x^4+x^3+x^2+x+1 $$ into quadratic and linear factors over $\...
0
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1answer
31 views

Formula For Rate of Increase of Money

If person 1 receives 1,000,000 dollars and wants to both keep 1,000,000 dollars and give 1,000,000 dollars to person 2 within 1 year, then the money must be doubled in 1 year. If person 1 wants to ...
1
vote
1answer
143 views

Prove that $(n!)^ 2 \gt n^n$ [duplicate]

Prove the above by by mathematical induction By any other method. I was just asked to prove this so I thought of using mathematical induction. My effort : I started first by verification and ...
1
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2answers
46 views

Property of a system of two inequalities

I have this system $$\begin{cases} a+b>1 \\ a-b>1 \end{cases}$$ can I sum the second inequality to the first getting $a>1$? Or this property can be used only equations?
0
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1answer
38 views

Prove that $1/a + 1/b - 1/c < 1/abc $, if $a^{2}+b^{2}+c^{2}=5/3 $and $a,b,c>0$.

I can't figure this one out. I got that $bc+ab-ab<1$. How do we apply the known sum of squares? How is this one solved?
4
votes
1answer
69 views

Find a function $f(x)$ such that $\forall \epsilon \gt 0, f(x) = f(x + \epsilon)$

Our professor asked us if we can find a function $f(x)$ such that $\forall \epsilon \gt 0, f(x) = f(x + \epsilon)$. In other words, a function that it's periodic no matter how small you pick the ...
1
vote
1answer
30 views

$\frac {\sin x - 1}{\sin x - 2}+\frac{1}2 \ge \frac {2-\sin x}{3 - \sin x}$

I've tried Cauchy, subtracting RHS from LHS, but can't solve it. Is there a simple way to do it?