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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1
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3answers
35 views

Proving $x^2 - y^2 + z^2 \gt (x - y + z)^2$

Prove that $$x^2 - y^2 + z^2 > (x - y + z)^2$$ where: $x < y <z$, $\forall x, y, z\in\mathbb R$ Thank for help.
5
votes
3answers
34 views

If $f(x) $ and g(x) are functions such that $f(x+y) =f(x)g(y) +g(x) f(y) $ then …

Question : If $f(x) $ and g(x) are functions such that $f(x+y) =f(x)g(y) +g(x) f(y) $ then $\begin{vmatrix} f(\alpha) & g(\alpha) & f(\alpha + \theta) \\ f(\beta) & g(\beta) & f(\beta ...
0
votes
0answers
15 views

Multi-ruled combinatorics problem (need this for my lab)

I need to know this for practical purposes and not homework, learning etc.. Say I have 3 electrodes A,B and C. Say I also have 3 electrolytes A,B and C. If electrode A has to be in electrolyte A, ...
3
votes
1answer
19 views

Function notation meaning: $f: \{a,b\} \to a$ - Zorich - MA I - p18

I have some notation I haven't seen before: $$f: \{a,b\} \to a\text{ and } g:\{a,b\}\to b$$ What does this mean? We are mapping from some $X=\{a,b\}$ to some $Y=a$? So pretty much we are always ...
1
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0answers
32 views

Which math class next

I just finished and Algebra for Calculus class this semester. I'm trying to work up to taking calculus (have to do up through calc 3). One person told me I should take trig next, and another calculus. ...
-2
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1answer
20 views

How is this problem solved? Steps? And what is the answer?

A company is planning to manufacture portable satellite radio players. The fixed monthly cost will be 300,000 and it will cost $10 to produce each player. a. Write the average cost function, C, of ...
-1
votes
1answer
27 views

How do you solve this word problem? And what is the answer? [on hold]

The data for the system's outflow can be modeled by the formula $$B = 0.07x^2 + 47.4x + 500$$ where $B$ represents the amount paid in benefits, in billions of dollars, $x$ years after 2004. According ...
0
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0answers
23 views

What is the point of reflection of this function

$$y = 3x(x+5)^{2/3}$$ Is there some kind of trick to simplify it?
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2answers
18 views

How can a given length of something yield different sum in square meters?

How can a rope of say 100 meters yield different return in square meters, based on how you divide each side? E.g. 10m x 10m = 100m2 15m x 5m = 75m2 Now of course I see that based on how you choose ...
1
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2answers
33 views

Math question related to Train

Two trains X and Y, leaves from point A and B towards B and A at the same time, after meeting each other they takes 4 hr 48 min and 3 hr 20 min to reach the point B and A. If the speed of train X is ...
1
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1answer
23 views

Graphing inequalities on a number line

What software or websites for graphing inequalities on a real number line?
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3answers
68 views

Is it possible to find [on hold]

If $$\frac {(a-b)(c-a)}{(b-c)(d-c)}=\frac {2012}{2013}$$ then find the value of $\dfrac {(a-c)(b-d)}{(a-b)(c-d)}$ in terms of numbers Note: $a,b,c,d$ are real numbers
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votes
1answer
41 views

An easy question regarding Algebra [on hold]

Three schools $A, B, C$ have a total of $480$ students. Ten per cent of the students of school $A$ are going camping and the percentages for school $B$ and school $C$ are 8.5% and 15% respectively. ...
1
vote
2answers
31 views

how to prove $a+b-ab \le 1$ if $a,b \in [0,1]$?

Given: $0 \le a \le 1$ $0 \le b \le 1$ Prove: $a + b - ab \le 1$
3
votes
3answers
49 views

Factorising quadratics - coefficient of $x^2$ is greater than $1$

In factoring quadratics where the coefficient of $x^2$ is greater than $1$, I use the grouping method where we multiply the coefficient and constant together and then factor. My question is can ...
0
votes
1answer
69 views

Questions about $f(n)=3+\frac{12}n$

Experimental Psychology: To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose the time in ...
6
votes
2answers
339 views

Quadratics with roots as integers; possible values of a

Suppose $a$, $b$ are real numbers such that $a+b=12$ and both roots of the equation $x^2+ax+b=0$ are integers. Determine all possible values of $a$. I don't know how to go about doing this without ...
1
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1answer
21 views

Remainders and polynomial division

Completley impromptu, one of my extended middle school students asked a question about her additional maths she was studying outside of school. For a certain polynomial, f(x), the remainder on ...
0
votes
1answer
29 views

Solving for the roots of a polynomial

Suppose we have a polynomial of the form: $$-x^3+3x^2+9x-27=0$$ Is there an easy way to find the solutions of $x$? I know that they will be factors of $27$, so I begin by factoring $27$ into ...
1
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2answers
24 views

Solving equations with exponentials and trig algebraically

Is it possible to algebraically solve an equation of the following form? $A\sin(x)+Be^x=C$ If so, how?
0
votes
2answers
43 views

Faster way to for $z^3 = -2 (1+i \sqrt 3) \bar z$ than complex algebra

What is the fastest way to solve for $z^3 = -2 (1+i \sqrt 3) \bar z$? I know how to do this using complex algebra. but that takes a long time. Can someone show me a faster way?
2
votes
2answers
39 views

Simplifying $\frac{\sqrt{3}}{2\sqrt{3}+1} + \frac{\sqrt{3}}{11}$

I realize that is a basic math problem, but I am still having problems with it. The expression $$\frac{\sqrt{3}}{2\sqrt{3}+1} + \frac{\sqrt{3}}{11}$$ equals one of the following: $2\sqrt{3}-1$ ...
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0answers
24 views

Let $X = \mathbb{R}$ and $Y = \left \{ x \in \mathbb{R}\mid x ≥ 1 \right \}$. Define $G : X → Y$ by $G(x) = e^{x^2}$. Prove that $G$ is onto. [duplicate]

Let $X = \mathbb{R}$ and $Y = \left \{ x \in \mathbb{R}\mid x ≥ 1 \right \}$. Define $G : X → Y$ by $G(x) = e^{x^2}$. Prove that $G$ is onto.
5
votes
1answer
54 views

no. of real solution of the equation $1+8^x+27^x = 2^x+12^x+9^x.$

The no. of real solution of the equation $1+8^x+27^x = 2^x+12^x+9^x.$ $\bf{My\; Try::}$ Let $2^x=a>0$ and $3^x=b>0\;,$ where $x\in \mathbb{R}$ So equation convert into $1+a^3+b^3 = a+a^2b+b^2$ ...
0
votes
2answers
30 views

Rearrange $y = xa-zc$ so that $a-c$ is on one side of the equation.

Is it possible to rearrange the following equation so that $a - c$ is on one side of the equation? $$ y = xa-zc $$ Thanks!
0
votes
1answer
7 views

Is every zonal homogeneous polynomial a polynomial on the unit sphere?

Let $$P_k(x_1\ldots x_n)=\sum_{\lvert \alpha\rvert=k} c_\alpha x_1^{\alpha_1}\ldots x_n^{\alpha_n}, \qquad (x_1\ldots x_n)\in \mathbb{R}^n$$ be a homogeneous polynomial of degree $k$. Assume that ...
2
votes
4answers
69 views

Why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$

I would like to understand why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$ I am particularly concerned with the term, $-4$.
-1
votes
2answers
24 views

Solving Inequalities with the use of their properties and cases [on hold]

Solve following inequality $$\dfrac4x + 3 \gt \dfrac2x + 1$$ and then graph the solution set on real number line.
5
votes
4answers
88 views

Prove that $13\vert(3^{n+1} +3^{n} +3^{n-1})$

Prove that $3^{n+1} +3^{n} +3^{n-1}$ is divisible by $13$ for all positive integral values of $n$
0
votes
2answers
22 views

Using the basic laws of exponent [on hold]

I have some problems with this question. Please help me. Thanks Simplify given expression$$ a^2 (abc)^{-2} a^3 b^7 $$ What are exponents of $a$, $b$, and $c$? I get $3,5,-2$ as exponents of ...
0
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2answers
18 views

Express the given expression as a single logarithm

Express $$2 \ln (2 - x) + 3 \ln (x^2 - 5)$$ as a single logarithm. Can anyone help me with this question? Thanks
1
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3answers
45 views

Show that $2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$

If $a,b,c$ are positive real numbers, not all equal, then prove that $$2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$$ How can I show this?
7
votes
3answers
66 views

valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
0
votes
1answer
50 views

Find exact value of $\sin\left(\dfrac x2\right) $

I have tried this problem over and over but can not get it. Can anyone provide a solution? Given $\sin(x) = -\dfrac67$ and $\tan(x)\gt0$ , find the exact value of $\sin\left(\dfrac x2\right) $.
0
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2answers
31 views

Solve and put in interval notation $4x^3 - 81x< 0$

The question is: Solve $4x^3 - 81x < 0$ and express the solution set in interval notation. I got $(-9/2,0)\cup(9/2,\infty)$ but I don't think its right. I factored it out to $x(2x+9)(2x+9)$
0
votes
3answers
65 views

Minimum value of $ f(x) = \frac{2+\sin x}{2+\cos x}$.

Minimum value of $\displaystyle f(x) = \frac{2+\sin x}{2+\cos x}$. My try: let $$\displaystyle y = \frac{2+\sin x}{2+\cos x}\Rightarrow 2y+y\cdot \cos x = 2+\sin x$$ So $$y\cdot \cos x-\sin x= ...
0
votes
2answers
48 views

If $\lim_{x\rightarrow \infty}\left[\left(x^5+7x^4+2\right)^c-x\right]$ is a finite, Then limit is

For a certain value of $'c',\lim_{x\rightarrow \infty}\left[\left(x^5+7x^4+2\right)^c-x\right]$ is a finite and non-zero, Then value of limit is $\bf{My\; Try::}$ Let $\displaystyle ...
1
vote
2answers
75 views

What is the angle that an Archimedean conical spiral makes with the floor?

I have a spiral in the form $$r = r_0(1-{\theta\over2\pi k }) \{r \ge 0\}$$ where $r_0$ is an initial radius, and $k$ is the number of turns. (It is a spiral that decays from $r_0$ to $0$ as $\theta$ ...
13
votes
2answers
884 views

Interesting Question on Ants

A horizontal stick is one metre long. Fifty ants are placed in random positions on the stick, pointing in random directions. The ants crawl head first along the stick, moving at one metre per minute. ...
0
votes
1answer
15 views

Unclear Application of Cauchy's Inequality

I was looking for a solution to a problem (both found here), where I came across the following ($a, b, c > 0$): Applying Cauchy's inequality, we get $(\frac{c}{a+2b} + \frac{a}{b+2c} + ...
0
votes
1answer
25 views

How can I solve this expression for x?

I would like to solve for $x$ given that \begin{equation} e^{-x}-\gamma-\eta e^{-\lambda(z-x)} = 0 \end{equation} where $\gamma, \eta, \lambda$ are positive constants and $z$ is a real number.
0
votes
1answer
25 views

questions related to progression [on hold]

Along a road lie an odd number of stones and distance between consecutive stones is 10m. A person can carry only one stone at a time and his job is to assemble all the stones around the middle stone. ...
0
votes
1answer
24 views

Giving a geometric representation of Cartesian products

What is being asked of me? Question 4 of Zorich(page 11) is exactly the following Give geometric representations of the following Cartesian products a) The Product of two line segments (a ...
0
votes
0answers
14 views

Is there a way to arrive at a funtion or a formula based on the outcome

The following table shows the input and the output. I'm trying to create a function that would relate the input and the output. SNU C020 C100 C300 C600 0 0 0 0 0 ...
3
votes
3answers
70 views

Solve: $\sin x - y\cos x = z$ for $x$.

I am working on programming a series of algorithms into a project, however I have run into trouble trying to solve this equation for $x$: $$ \sin x - y\cos x = z $$ It should be noted that $y$ and ...
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3answers
44 views

Best argument to prove $|x|\le a \iff -a\le x \le a$

$$|x|\le a \iff -a\le x \le a$$ I can only verify the integrity of this by talking about distances on the number line. But is there a algebraic argument that proves this?
0
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0answers
10 views

Locus of intersection between $y= 8\lambda/(\lambda ^2 + 4)$ and $y =2 \lambda x/(4-\lambda^2)$

I have the equations $$y=\frac{4\lambda}{\frac{1}{2}\lambda^2+2}\quad \text{and}\quad y=\frac{\lambda x}{-\frac{1}{2}\lambda ^2 + 2}$$ each representing a line. I'm asked to find the locus of the ...
0
votes
3answers
49 views

simplifying $-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$

simplifying $-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$ in my lecture notes somehow my lecture got from$-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$ to ...
0
votes
0answers
40 views

Neutron-Density cross-plot interpretation

I have a question about solving a particular graphical problem. This is a picture of a Neutron-Density cross-plot: It's a little bit confusing as plots go, so allow me to try to explain the salient ...
-1
votes
0answers
22 views

Problem about a focal chord

Given parabola $y^2=4ax$ with length of the focal chord equal to $l$ and the length of the perpendicular from vertex to the chord is $p$. Which one of these statements is true? 1) $l⋅p$ is constant ...