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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15 views

How do I find the domain and range of this piecewise defined function?

Are both conditions true when $x>3$? If so, how do I graph it? $$ f(x)=\left\{\begin{aligned} &x^2-4&&:x>3\\ &2x-1&&:x\geq 3 \end{aligned} \right. $$
0
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2answers
26 views

Why does $1-\frac{1+n}{4+2n}=\frac{3+n}{4+2n}$

I have an understanding of proofs and I came to the point where $1-\frac{1+n}{4+2n}=\frac{3+n}{4+2n}$. I couldn't figure out why the math worked out though. Could someone explain to my why this math ...
0
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4answers
45 views

Help understanding how to factor completely $x^3-x^2-x+1$

I need someone to help explain the steps to completely factor the problem $x^3-x^2-x+1$. Here is what I have done so far: $x^3-x^2-x+1$ to $x^3-x^2+-1(x+1)$ Since there is a ...
0
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3answers
50 views

Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ be unit vectors, with $\mathbf{a+b+c=0}$. The angle between any two of these vectors is $120^\circ$.

Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ be unit vectors, such that $\mathbf{a}+\mathbf{b}+\mathbf{c} = \mathbf{0}$. Show that the angle between any two of these vectors is $120^\circ$. Hi, ...
0
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1answer
28 views

Show that $f(a)$ converges after some point

There is a row of 1000 integers. There is a second row below, which is constructed as follows. Under each number $a$ of the first row, there is a positive integer $f(a)$ such that $f (a)$ equals ...
0
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0answers
16 views

Algebra inequality problem

Suppose: $ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$ , and $x_1x_3x_5 + x_2x_4x_6 \ge \dfrac {1}{540} $ and $\dfrac{p}{q}$ is the maximum possible value of $x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 ...
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1answer
28 views

Anna, Bill and Carl entered a race [on hold]

Anna, Bill and Carl entered a race. Bill's speed was 4/5 Anna's, and Carl's speed was 3/4 Bill's. How many times the mean of the two boys' speeds was Anna's?
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0answers
17 views

Polynomial division of $x^3-y^3$ using lexical ordering $x > y$

I am trying to divide $f = x^3 - y^3$ with $f_1 = x - y$ as the divisor using lexical ordering $x > y$. I know that $LT(f) = x^3$ and $LT(f_1) = x$, so $f_2 = f - x^2 f_1 = x^2 y -y^3$. So, ...
0
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3answers
45 views

Solving $\frac{x+y}{xy}=2$, $\frac{x-y}{xy}=6$

$$\frac{x+y}{xy}=2,\ \ \frac{x-y}{xy}=6$$ I am not understanding how to solve the equation. I tried diving the whole equation by $xy$, but, that didn't work too. Any hint or help would be much ...
1
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1answer
17 views

Finding out the logarithmic function for the situation below

The situation reads as follows: There are 3000 barbs in a pond and every year 20% of the barbs die and then 1000 new barbs come to the pond. A logarithmic function needs to be plotted to graph ...
0
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0answers
14 views

how to find principal when rate and sum of compound interest is given? [on hold]

Find the money, invested at 10% compounded annually, on which the sum of interest for first year and third year is 1768
-1
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1answer
41 views

tricky inside/outside of brackets in algebra

Oh hi guys I'm hoping to get some help with these two questions. I know the basic rule about brackets, but I'm stumped here: a) $8P - (3 + P) = 12$ b) $28 - (1 - P) = 42$ Many thanks :d
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0answers
15 views

How to determine when this two variable transformation is invertible?

I am given: $$ X= U \cos(V) \tag{1}\\ $$ $$ Y = U \sin V \tag{2}$$ Now, I need to: a) Give the respective ranges for $U$ and $V$ in order that the transformation defined is one to one. and ...
10
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3answers
114 views

What is the connection between the discriminant of a quadratic and the distance formula?

The $x$-coordinate of the center of a parabola $ax^2 + bx + c$ is $$-\frac{b}{2a}$$ If we look at the quadratic formula $$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ we can see that it specifies two ...
1
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1answer
12 views

When can you not do a mapping composition?

Suppose I have $\alpha:\mathbb R^3 \to \mathbb R$ and $\beta:\mathbb R \to \mathbb R^+$. Looking over my notes, it says $\alpha \circ \beta$ can not be done but $\beta \circ \alpha$ can. What is the ...
3
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2answers
40 views

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$ What I did : Use $g(x) = f(x)-x$, then $g'(x) = f'(x)-1\ne0$ I suspect I have to use Rolle's theorem now, But I am having difficulty ...
1
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4answers
25 views

Solution set of modulus inequations

$$|2x+5|\leq\dfrac{1}{2}$$ What will be the solution set? My attempt: For $x \in \left(-\infty,-\dfrac{5}{2}\right)$ $x\geq -\dfrac{11}{4}$ For $x \in \left[-\dfrac{5}{2},\infty\right)$ $x\leq ...
0
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3answers
43 views

How does $\log_2(A)-\log_2(B)+\log_2(c)$ not equal $\log_2(\frac{Bc}{A})$

$\log_2(A)-\log_2(B)+\log_2(c)$ How does this equal $\log_2(\frac{Ac}{B})$? Does it not follow from the order of operations that it would be addition first then subtraction? I'm having hard time ...
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2answers
32 views

Need help understanding how to simply and combine this problem.

I'm sorry to bug the community with such a simple problem but finals are near and my professor is MIA. I was able to copy down part of the solution, but I don't understand how he got to that part, and ...
3
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1answer
34 views

Why an exponential graph can't have b equal to 1

I've seen that the graph of an exponential function, $f(x) = a\cdot b^x$, cannot have $b$ equal $1$. Why is this? I think it's because the function would be a flat line if $b=1$. Is this true?
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3answers
37 views

Finding the length of a side of a triangle

I just took the SAT and was wondering if there is any way to find out the length of a side of triangle when you know the three angles and the area of the triangle.
1
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1answer
34 views

Maximum value of $\,{f(x) = \min (3x + 2, 5x + 1)}$

Given that $f(x) = \min (3x + 2, 5x + 1)$, what is the maximum possible value of $f(x)$? Both $3x+2$ and $5x+1$ are increasing functions and irrespective of which expression $f(x)$ takes, one thing ...
2
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3answers
46 views

Finding sum of real roots when when higher roots given

What are the sum of real roots of the equation $(2+(2+(2+x)^2)^2)^2=2000$ i tried taking square root of both sides but couldn't come up with something. Any help is greatly appreciated! (Turkish ...
0
votes
1answer
16 views

Simple equation solving with two variables

This is my first post here, so bear with me. Also, my first language isn't English, but I hope that won't be a problem. So my question is probably a piece of cake, but since I'm a youngster and ...
1
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3answers
129 views

I need help identifying the slope for an equation

I'm currently having a problem with this problem ---> 2x – 5y = 10 Im supposed to Write given equation in slope-intercept form and identify slope: I did that, but I thought the answer was 5, but it ...
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0answers
24 views

Algebra word problem - question on “man-hours” [on hold]

Working together, two workmen take 14 hours to complete a job. How much time would it take them to complete it separately if one of the workmen is two times faster than the other? I need the answer ...
1
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1answer
28 views

Grid Problem Proof

I have a 2x2 grid square say, I can fit a shape like this: Such that there is one missing square. I can arrange this in any way so that the missing square can be located anywhere. I can do ...
13
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2answers
795 views

Is Vieta the only way out?

Let $a,b,c$ are the three roots of the equation $x^3-x-1=0$. Then find the equation whose roots are $\frac{1+a}{1-a}$,$\frac{1+b}{1-b}$,$\frac{1+c}{1-c}$. The only solution I could think of is by ...
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4answers
37 views

Can you help me subtract intervals?

I was reading my abstract math textbook and they subtracted $[3, 6] - [4, 8) = [3, 4)$. I was wondering if someone could write out how they got to $[3, 4$). I looked at wikipedia and it said I should ...
3
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4answers
117 views

Find the sum of values of $x$ such that $|x+2| +|x-3| +|x+4| + |x+5| = 18$

I tried it by finding the different values at the $4$ inflection points of the graph. Then didn't know how to proceed. Am I correct till here?
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0answers
10 views

Proving that equilateral triangle has equal medians. [on hold]

How to prove that equilateral traingle has equal medians? Mathematical method. Thank you. :D
2
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0answers
26 views

A question on numbers satisfying $|x − a| = k |x|$ [on hold]

Let $x, a$ and $k$ be positive reals. If $x \in [0,a]$ satisfies $|x − a| = k |x|$ then $x$ divides the interval $[0,a]$ in what ratio? I need help with this question, I don't understand what to ...
0
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4answers
30 views

Mixed Fractional Equation?

$$3 \frac{3}{5} + \frac{2}{x} = 4\frac{4}{15}$$ I tried subtracting by both sides, etc, but it didn't come out right. I also tried multiplying by both sides, but, it didn't seem to work. what would ...
1
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1answer
25 views

Factor polynomial with irrational roots using quadratic equation

If I want to factor the polynomial $x^2 + 3x + 1$, I thought I could use the quadratic formula to find that its roots are $\dfrac{-3\pm\sqrt{5}}{2}$. Then, since those are both negative values, take ...
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1answer
18 views

Calculating AER with an unknown.

I got this word problem that I can't figure out. It's pretty simple, but I can't seem to get it right no matter what I try. If, instead, the \$4000 was invested in a 3-year savings bond which also ...
6
votes
1answer
99 views

How to prove that $a=z^{p}$ for some $z \in \mathbb{Z_{+}}$?

Claim : If for a positive, composite integer $a$ and an odd prime $p$, such that $\gcd(a,p)=1$, we are given $$ a^{p^{n-2}(p-1)} \equiv 1 \pmod {p^n} \ \forall \ n \geq 2 \ \ ;\ ...
3
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4answers
32 views

Finding the sum of real solutions to an equation

how to find the sum of real solutions if, $(x+1)(x+\frac14)(x+\frac12)(x+\frac34)=\frac{45}{32}$ I have tried multiplying both sides with 32 and got ...
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votes
1answer
35 views

Finding biggest real root on an equation [on hold]

What is the biggest real root of $x^3-x^2-x-\frac{1}{3}=0$ a) $\frac{\sqrt{3}-\sqrt{2}}{2}$ b) $\frac{\sqrt[3]{3}-\sqrt[3]{2}}{2}$ c) $\frac{1}{\sqrt[3]{3}-1}$ d) $\frac{1}{\sqrt[3]{4}-1}$ e) None ...
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0answers
21 views

driving question in a speed zone [on hold]

If you are driving 35mph and the photo radar machine is $\frac{1}{10}th$ of a mile away, and at that point the speed limit is $15mph$, is the radar placed at an unfair point??
2
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2answers
103 views

What is the value of $\frac{a^3}{a^6+a^5+a^4+a^3+a^2+a+1}$

If $\frac{a}{a^{2}+1} = \frac{1}{3}$ Then find the value of $$\frac{a^3}{a^6+a^5+a^4+a^3+a^2+a+1}$$ Any hints to help me?
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4answers
49 views

How find the fractional part of $5^{200}$ divided by $8$?

Finding the fractional part of $\frac{5^{200}}{8}$. I've had this problem given to me (we're learning the Binomial Theorem and all.) So obviously I thought I'd apply the binomial theorem to it, ...
0
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1answer
19 views

Finding the length of a train confusion

Erica noted that a train to Muizenberg took $8$ minutes to pass her. A train in the opposite direction to Cape Town took $12$ minutes to pass her. The trains took $9 $ minutes to pass each other. ...
5
votes
5answers
127 views

Is it possible that $(f\circ g)(x)=x$ and $(g\circ f)(x)\ne x$?

Is it possible that $(f\circ g)(x)=x$ and $(g\circ f)(x)\ne x$ In other words, To show $f$ and $g$ are inverse, is it enough to show $(f\text{ o }g)(x)=x$? I have never witnessed a case in which the ...
1
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2answers
37 views

Equations with Sinus. How to find equation solution?

I don't understand how to get a solution for sinus equation. I have: $$\sin x = \frac{1}{2}$$ \begin{align*} x & = (-1)^k \cdot \arcsin\left(\frac{1}{2}\right) + 180^\circ \cdot k, k \in ...
0
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1answer
19 views

Difference between orthogonal complement and Gram-Schmidt process

Could someone explain what is the difference between these two? From my poor understanding they seem to do the same thing, given a set of vectors we find their corresponding orthogonal vectors. Maybe ...
0
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1answer
30 views

Exist another method to solve the problem?

We have $x_1,\:x_2,\:x_3\:\in \:\mathbb{C},\:\:f=x^3+x^2+mx+m,\:m\in \mathbb{R}$. We need to find $m\in\mathbb{R}$ such that $|x_1|=|x_2|=|x_3|$. Here is what I tried: $f=x^3+x^2+mx+m=(x^2+m)(x+1)$, ...
0
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2answers
34 views

What are the steps to compute $\sqrt{x^2+1}$ to the form $x\sqrt{1+ \frac{1}{x^2}}$.

How can someone compute step by step the function $\sqrt{x^2+1}$ to the form $|x|\sqrt{1+ \frac{1}{x^2}}$? The only way I know how to take something out of a root is to factor inside the root and if ...
-1
votes
1answer
19 views

Geometric sequences with money [on hold]

Use Geometric sequences to answer the following question: Which of the following will net you more money. $100,000 a day or doubling what you were given the previous day starting with 1 cent if this ...
-4
votes
0answers
15 views

Average Number to the nearest tenth [on hold]

A school has 112 homeroom classes. There were 15 school class in February. The aggregate register of the school for the month of February was 52, 920, the aggregate attendance was 43, 860 the average ...
-4
votes
2answers
30 views

What is the value of x when Rectangle JKLM is shown [on hold]

What is the value of x when Rectangle JKLM is shown Rectangle JKLM is shown. http://education.jlab.org/solquiz/img/7261.gif Rectangle JKLM f 12 g 29 h 43 j 28