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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31 views

Any idea for proving this equality?

I just can't come up with any idea for proving that $x=\alpha$, where: $x=aA_1+(1-a)A_2$, $a=\frac{A_1}{A}$, $A=F_1$+$F_2$, $A_1=\frac{f_1}{F_1}$, $A_2=\frac{f_2}{F_2}$, ...
0
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1answer
44 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
35 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 $y$?
0
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0answers
25 views

What is the intersection of thses equivalence relations?

Let $S$ be the following subset of the plane: $$ S \colon= \{ \ (x,y) \ | \ y=x+1, \ 0 < x < 2 \ \}.$$ Then how to describe the equivalence relation $T$ on the real line that is the ...
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1answer
34 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 ...
0
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1answer
35 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 ...
0
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4answers
54 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.
1
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1answer
28 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
56 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 + ...
0
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2answers
28 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 ...
2
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2answers
21 views

What conclusion could be drawn about the maps from their compostie?

Let $f \colon A \to B$ and $g \colon B \to C$. Then If $g \circ f$ is injective, then I know that $f$ is injective, but what can we say about the injectivity of $g$? If $g \circ f$ is surjective, ...
0
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2answers
26 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
46 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 $$\sum_k^n P_k \sum_k^nx_k$$
0
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0answers
34 views

Calculation of Minimum value of $f(x) = x^6+x^4-x^3-x+1$.

Calculation of Minimum value of $f(x) = x^6+x^4-x^3-x+1$. $\bf{My\; Try::}$ $\bullet \; $If $x\leq 0\;,$ Then $f(x) = x^6+x^4+(-x)^3+(-x)+1\geq 1$ $\bullet \;$ If $x\geq 1\;,$ Then ...
0
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3answers
21 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 ...
1
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1answer
90 views

Cyclic sum inequality. $x^2+y^2+z^2=3/2$

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 ...
0
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2answers
70 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
51 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
45 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
36 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
30 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
votes
3answers
161 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) = .. ...
2
votes
4answers
131 views

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

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
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1answer
53 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
votes
1answer
192 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 $$ ...
10
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2answers
179 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
65 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
35 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: ...
0
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3answers
52 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
55 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.
1
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2answers
43 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
vote
2answers
65 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
votes
0answers
53 views

What is the sum of 1^2 + 2^2 + 3^2… + T^2? [duplicate]

I need to find the sum of this sequence. Thank you in advance
0
votes
1answer
19 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
107 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
vote
2answers
36 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
26 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
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1answer
59 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
28 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?
-1
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1answer
31 views

Trouble in solving an equation [closed]

Hi i'm having trouble solving this equation, it would be very appreciated if you could help me out: $(9-4d^2)(9-d^2)=385$ Please explain
3
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1answer
59 views

Why doesn't this method of solution work?

Solve $$\sqrt{2x^2 - 7x + 1} - \sqrt{2x^2 - 9x + 4} = 1 \tag1$$ I tried to do the following: $$(2x^2 - 7x + 1) - (2x^2 - 9x + 4) = 2x-3\tag2$$ Dividing $(2)$ by $(1)$ yields $$\sqrt{2x^2 ...
3
votes
4answers
165 views

What is $(-8)^\frac{2}{3}$?

I am comfuse about something. I want to compute $(-8)^\frac{2}{3}$ Is it $(-2^3)^\frac{2}{3}$=$(-2)^{3\cdot\frac{2}{3}}$=$(-2)^2=4$ ? Is there any problem here because the base is negative? ...
0
votes
1answer
18 views

Formula For Finding the Next Near Consecutive Perfect Square

For any three consecutive members of a sequence, the first and third members are near consecutive. 1 squared is 1. 2 squared is 4. So 1 and 4 are consecutive perfect squares. 1 squared is 1. 3 ...
2
votes
2answers
33 views

Limit at negative infinity

Evaluate $$\lim_{x\to -\infty}\frac{\sqrt[3]{x^6+8}}{4x^2+\sqrt{3x^4+1}}$$ I think the strategy is to divide the numerator and denominator by x^2. Help please. The textbook answer is ...
0
votes
1answer
27 views

Semi-log re-expression to find an exponential model.

I'm unsure of how to approach this problem because of the 'semi-log' part, would I find the line of best fit, then log both sides on that equation until it is in exponential form? Thanks in advance. ...
0
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2answers
29 views

Possible to solve for a common base number?

This may be a little low-brow for this forum, but I'm trying to figure out what the common base number set is between two other sets of numbers. Here's the situation: I have received quotes from two ...
0
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0answers
27 views

Help with this binomial coefficent problem

find the binomial coefficient of $(1+t+t^3)^n$ I know the binomial coefficient, as i was working on it i end up having double summations and i don't know if that is even possible. I feel like there ...
0
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1answer
34 views

Buying dogs and Cats and mice

This problem is only considering positive integer solutions: You must spend exactly 100 and purchase exactly 100 animals. Each dog costs 15 and each cat costs 1 and each mouse costs .25. How many of ...
0
votes
1answer
71 views

Simplifying $2^\sqrt{\log x}$

Can this expression be simplified? $$2^\sqrt{\log x}$$ Thank you
0
votes
1answer
23 views

Is there a general formula for the following expression?

I am working on a proof, and came to the point where I need a general expression for the formula, taking real numbers $x_i$ and an integer $k$, of : $(x_1+x_2+...+x_n)^k$ I now I could apply the ...