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
57 views

Variance of Portfolio Return

I am struggling to understand the following proof. I do not understand the highlighted red step. Please could this be broken down so that I can understand it. It seems like a manipulation of ...
1
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3answers
324 views

Finding all the set of values for k such that $-2x^3+24x+8=k$

find all the set of values for k such that $-2x^3+24x+8=k$ has more than one solution. This problem can only be solved using basic Calculus 1 knowledge. I thought about setting the equation to zero ...
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1answer
26 views

Proving absolute inequality

Suppose $b$ is an integer and $r_1$ and $r_2$ are positive integers such that: $0 < r_1 < |b|$ and $0 < r_2 < |b|$. Prove that : $|r_1 - r_2| < |b|$
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1answer
66 views

How find the equaition $x^2\sin{\dfrac{1}{x}}=2x-501$ root

find the equation approximate solution , such the root of $$x^2\sin{\dfrac{1}{x}}=2x-501$$ to an accuracy of $ 0.001$ I think this problem use this $$\sin{x}\approx x-\dfrac{1}{6}x^3$$ ...
2
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1answer
618 views

Exponential Function Shifts

I have some confusion about shifts concerning exponential functions. I can best describe my question with an example. Take $y = e^{-(x-3)}$. This graph has a reflection over the $y$-axis and is ...
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1answer
71 views

Representing a a quantity as a mathematical expression

Each parent in a group of 15 parents buys a ticket to a fundraiser show and receives a group discount of $2 off of the regular ticket price. I need to write an expression that represents the total ...
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1answer
85 views

Systems of equations question

\begin{align*}a^a\cdot b^b\cdot c^c\cdot d^d&=\frac12\\a+b+c+d&=1\end{align*} How can we find solutions for this system of equations given that $a, b, c, d > 0$ ?
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1answer
50 views

How do I prove the existence of x that satisfies this inequality?

I'm trying to solve this inequality: $(a+b+c+ab+ac+bc+abc)/7 > x > abc > 1$, where $a,b,c > 0$ and real-numbered, with the caveat: x is supposed to be a member of this set ...
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0answers
35 views

Sequence of intersecting lines

Suppose we have the following: The lines of $y=3x+8$ and $3x+y+4=0$ meet at point $A$. Line $L$ goes through point $A$ and point $(7, 11)$. Line $L$ meets the $x$ axis at point $B$. Line $K$ goes ...
2
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3answers
109 views

Taking Radicals Out of Denominator

How does one take radicals out of the denominator of $$\frac{1}{a + b(\sqrt{2} + \sqrt{3})}$$ assuming $a,b \in \mathbb{Q}$ not both equal to $0$. I know the trick is to multiply this expression by ...
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2answers
68 views

Smallest value of $x^4+2x^2+y^4-2y^2+3$

The question asks: Find the smallest value (for real $x$ and $y$) of: $$x^4+2x^2+y^4-2y^2+3$$ I don't think I understand this question, it is in a completing the square exercise and I don't really ...
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2answers
109 views

What is the best response to the reasoning below? [closed]

I would like some input into this question I need to solve. From my logic: $a = 1$. we get this from $1=2$ ($a=2a$). It appears $b = 0$. because $a = a + b$ ($1=1+0$). so in the original statement ...
1
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1answer
168 views

Explanation on a double summation

Hello: I was wondering if someone will help me understand what's going on in this summation. I don't think the terms are really important. I'm looking for an explanation why the left hand side is ...
0
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1answer
93 views

Why is only the first (highest) term of the divisor in polynomial long division used to divide?

There is one small matter that has always stumped me with polynomial long division. In the example from the Wikipedia on Polynomial long division, why is the equation only divided by the first/highest ...
1
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1answer
37 views

number of images from $A$ to $B$

Let $A = \{2,4,6,8,10,12\}$ and $B = \{3,7,11\}$. Then no. of function such that element $3$ of $B$ has two , $7$ of $B$ has three and $11$ of $B$ has one pre - image in $B$ ,is My Try:: for ...
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1answer
95 views

find the values of x

I am trying to understand how to do this problem. Can someone guide me please? Here is the problem: ...
0
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2answers
517 views

Pentagon axis of symmetry

Quick question: Can a pentagon have an axis of symmetry passing through two (one, none) of its vertices? I'm given the following definition for axis of symmetry: A figure is said to have an axis of ...
3
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1answer
182 views

Solve the simultaneous equations $x + \frac{3x-y}{x^2+y^2} = 3 $, $ y – \frac{x+3y}{x^2+y^2} = 0$

Find all solution in $\mathbb{R}$ for the following system of equations: \begin{cases} x + \frac{3x-y}{x^2+y^2} = 3 \\ y – \frac{x+3y}{x^2+y^2} = 0 \end{cases} I've tried few method, but none ...
2
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1answer
103 views

Does such a function exist: $\left|f(x) - f(x+1)\right| \ge {1\over x+1}$?

Does a function $f(x)$ exist such that $\left|f(x) - f(x+1)\right| \ge {1\over x+1}$ where the range of the function does not extend infinitely in the positive direction. If not why?
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2answers
207 views

How to graphically solve this inequality

Could anybody show me how to solve this ? $$|y| \leq |x-1|$$
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3answers
49 views

Simplification (algebra)

The problem states just to simplify: $$ \frac{(x+1)^2(x-2)-(x+1)(x-2)^2}{(x-2)^3} $$ I was given the answer as $3(x+1)/(x-2)^2$, but every time I try to work it out I get a different answer.
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2answers
318 views

If 10% of a substance decays in 10 years, what is the half-life of the substance?

A radioactive substance decays at a rate proportional to the amount present. If 10% of such a substance decays in 10 years, what is the half-life of the substance?
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2answers
296 views

Difference Quotient for $\;\frac 1{x^2}$

It's been a while since I've done this kind of math and I've been asked to help explain this concept to a high school student. Given $$g(x) = {1\over x^2}$$ we are asked to find the difference ...
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2answers
75 views

How $\alpha(\alpha+1)\ldots(\alpha+k-1)=\frac{\Gamma(\alpha+k)}{\Gamma(\alpha)}$?

Probability function of Negative Binomial Distribution, $NB(\alpha,p)$, is $$P(X=k)=\binom{\alpha+k-1}{k}(1-p)^{\alpha}p^k,\quad \alpha>0$$ Probability generating function of Negative Binomial ...
1
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1answer
339 views

How do I find vertical asymptotes of this function?

$$ f(x) = 4\tan(\pi x) $$ I am having trouble finding the asymptotes for this problem. From what I gathered the asymptotes of $\tan(x)$ occur at $x=\frac{\pi}{2} +n\pi$. How would I solve this using ...
4
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1answer
343 views

Why can't I take the square root of both sides when finding the roots of a quadratic equation?

Is $x^2 - x - 12 = 0$ equivalent to $x = \sqrt{x + 12}$? I started with $x^2 - x - 12 = 0$ and made the following changes: $x^2 - x - 12 = 0$ $x^2 = x + 12$ $x = \sqrt{x + 12}$ From here I can ...
1
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2answers
256 views

Equation for determining a car's fuel consumption as well as cost

Purchase price: 24000 Avg km/year: 40000 L/100 km: 5.3 Price of gas (per L):1.30 I was wondering what the formula is to find out how much litres of gas the car would consume as well as the cost of ...
0
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1answer
161 views

Function and dependent variable are represented by the same symbol?

Is it wrong to represent a dependent variable and a function using the same symbol? For example, can we write the parametric equations of a curve in xy-plane as $x=x(t)$ , $y=y(t)$ where $t$ is the ...
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3answers
138 views

Calculate $(2013)^4 - 4(2011)^4+6(2009)^4-4(2007)^4+(2005)^4 = $

Calculate $(2013)^4 - 4(2011)^4+6(2009)^4-4(2007)^4+(2005)^4 = $ Try:: Let $x = 2009$, Then expression convert into $(x+4)^4-4(x+2)^4+6x^4-4(x-2)^4+(x-4)^4$ ...
3
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3answers
333 views

Find $(a,b)$ if $x^2-bx+a = 0, x^2-ax+b = 0$ both have distinct positive integers roots

If $x^2-bx+a = 0$ and $x^2-ax+b = 0$ both have distinct positive integers roots, then $(a,b)$ is $______$? My Try:: for first :: $\displaystyle x^2-ax+b = 0\Rightarrow x = \frac{a\pm ...
3
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1answer
84 views

Find the number $k$

Find the number $k$ such that the function $$f(x)=\sqrt{kx-9}$$ has domain $[1/2,\infty)$ and then find its inverse $f^{−1}$, and $f^{−1}(\sqrt2)$. Tried to find the $k$ by setting $\displaystyle ...
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1answer
48 views

Functions and graphs

I have been confusing about the graphs of functions since two or three months. I had searched in internet but could not get something. I want to know how the graphs of polynomials behave. For example, ...
2
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1answer
54 views

Algebra problem - length of vector

Algebra - Complex Numbers Calculate the length of the vector given by $\frac{\left( 1+2i \right)\left( 1+\sqrt{3}\cdot i \right)}{\left( 1+i \right)^{3}} $ I start by doing $\left( 1+i ...
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2answers
116 views

Manipulating surds and exponentiation

Sir, Can you tell me if my below simple calculation is right for -1/2 root of 1/2? $$\sqrt[-1/2]{ 1/2} = (-1/2)^2 = 1/4$$ I got this doubt because when I am trying this in http://web2.0calc.com/ , ...
3
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2answers
122 views

Solving an equation with absolute values inside absolute value

How do I solve this? $$\big||2x-6|-|x-9|\big|<3$$ I know you can just square $[()^2]$ the whole inequation, but in this case it becomes a very ugly expression. I'm sure there's an easier way!
1
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1answer
418 views

using the same symbol for dependent variable and function?

Is it wrong to represent a dependent variable and a function using the same symbol? For example, can we write the parametric equations of a curve in xy-plane as $x=x(t)$, $y=y(t)$ where $t$ is the ...
3
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3answers
69 views

unknown equals constant or constant equals unknown?

Frequently, in solving a simple equation involving an unknown such as x + 1 = 2, at the end, we often write the answer as x = 1 instead of writing the answer as 1 = x, that is an unknown equals a ...
0
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1answer
44 views

Having trouble with variable multiplied with a logarithm.

I have tried solving the following equation. $$f(N)=(N^{1/3})*(log_{2} N)^{2/3}$$ I guess I need to clarify my question. I need to solve for N somehow. For instance, if f(N) = 256, then what would N ...
0
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3answers
1k views

Sin properties - Why is this undefined?

Why is sin(-π/2)^(1/2) undefined? sin(-π/2) is -1, so I thought the the answer would be the same, -1^(1/2)=-1? But my calculator and the answers sheet both say undefined...
2
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5answers
916 views

Distance between a point and a plane

I was just working on some review textbook problems in James Stewart's Multivariable Calculus when I encountered a problem that looked like the following: Find the distance between the point ...
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1answer
31 views

Help with inequality question based on the exponential series

Here is a very old Year 12 (high school) exam question I can't quite figure out. It has three parts and I'm stuck on the last part. This is for my maths revision only: 1) write down the series for ...
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5answers
463 views

How to calculate $2^{\sqrt{2}}$ by hand efficiently?

I've been trying to calculate $2^{\sqrt{2}}$ by hand efficiently, but whatever I've tried to do so far fails at some point because I need to use many decimals of $\sqrt{2}$ or $\log(2)$ to get a ...
1
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2answers
1k views

How do I find all points on the y-axis that are a distance 6 from P(5,3)? [closed]

How do I solve this problem? find all points on the y-axis that are a distance 6 from P(5,3)
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4answers
126 views

Sum of series proof

How can I find the sum of $\sum_{i=1}^{n} i2^i$ in a closed form. I have a feeling I can do this using differentiation of an existing series but cant seem to find it.
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2answers
250 views

How to find a formula for this bijection?

We know that there is a bijection of the set $S:= \{ 2^m 3^n \mid m, n \in\mathbb Z, m,n\geq 0\}$ onto the set $\mathbb N$ of natural numbers. How to find a simple formula for such a map?
2
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3answers
101 views

If $\,x-\frac 1 x=k, \, k$ being any integer,then $\,\,x^5-\frac {1}{x^5}=?$

I am stuck with the following problem which one of friends gave me : If $\,x-\frac 1 x=k, \, k$ being any integer,then $\,\,x^5-\frac {1}{x^5}=?$ The options are $\,\,k^5+4k^3+4k, ...
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2answers
270 views

How to get all solutions to equations with square roots

I would like to find all solutions to $$b-a\sqrt{1+a^2+b^2}=a^2(ab-\sqrt{1+a^2+b^2})$$ $$a-b\sqrt{1+a^2+b^2}=b^2(ab-\sqrt{1+a^2+b^2})$$ I found some solutions. For example, $a = 1, b = \pm i$ and ...
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3answers
196 views

Given 49a + 49b = 6272, the average of a and b is 64. Why?

I found this math test very interesting. I would like to know how the answer is being calculate?
1
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1answer
138 views

induction (sum of squares of products of elements of certain subsets of $\{1,\dots,n\}$)

Let $n$ be any natural number. Consider all nonempty subsets of the set {$1,2,...,n$}, which do not contain any neighboring elements. Prove that the sum of the squares of the products of all numbers ...
0
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2answers
52 views

Simplifying two fractions on top of a third fraction

How would I go about simplifying this fraction: $$\frac{\frac{1}{x} - \frac{1}{a}}{x - a}$$ I've looked at similar questions such as this one but still can't seem to figure this one out. Any help ...