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
51 views

If $f(x)=2x-5$, when $f(x)=13$

If $ \ f(x) = 2x - 5 \ $ , find $ \ x \ $ when $ \ f(x) = 13 \ $. I substituted 13 and got 21, but the answer was 9? Can someone please show me working out for this type of question I'm bad at math!
0
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2answers
60 views

Trignometric Equation

Find the three smallest positive values of $ \theta $ such that $ 4\cos^2(2\theta-\pi) =3. $ I saw that $\cos{(2\theta-\pi)}$ equals $\frac{\sqrt{3}}{2}$, therefore, $2\theta-\pi=\frac{\pi}{6}+2x\pi$ ...
10
votes
4answers
1k views

The 'sine and cosine theorem' - formulas for the sum and difference

I've read somewhere that the sine and cosine functions can be fully described by this theorem: $\sin(0) = 0, \cos(0) = 1$ $\sin(a-b) = \sin(a)\cos(b) - \sin(b)\cos(a)$ $\cos(a-b) = \cos(a)\cos(b) + \...
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1answer
27 views

Help with finding equation?

Find the hyperbolic equation that satisfies: Foci:$(\pm3,0)$ and hyperbola passes through the point $(4,1)$. I have tried to say $3^2=a^2+b^2$, so $b^2=9-a^2$ and I know that $$\frac{x^2}{a^2}-\frac{...
0
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5answers
59 views

What are the values of $K$?

The values of $K$ for which the system \begin{cases} x + y = 5\\ y + z = 4\\ x + 2Kz = 1 \end{cases} has only one solution are… Will the answer be ( All Real Numbers ) or ( the empty set )?
1
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2answers
88 views

Function application (word problem)

The problem: My work so far: $3=log(\frac{A}{A_0})$--->$10^3=\frac{A}{A_0}$ $\frac{A}{A_0}=1000$ (Am I done there?) Plugging it in: $M=log(\frac{1900000}{1000})$ $10^M = \frac{1900000}{1000}$ ...
1
vote
2answers
50 views

How to solve $\dfrac{7x}{8}+4-\dfrac{2x}{3}=4x-3$?

$$\frac{7x}{8}+4-\frac{2x}{3}=4x-3$$ I do not understand how to simplify this. Could anyone here help me, please? Thanks.
0
votes
1answer
77 views

$.9999\dots = 1 $ What's wrong with this Algebra? [duplicate]

This just feels wrong. My professor is claiming that .999... = 1 with the following proof: $x = .999\dots$ $10x = 9.999\dots$ $9x = 10x - x = 9.999\dots = 9$ $x = 1$ Is this actually correct and ...
1
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0answers
159 views

Absolute values nested multiple times

Is there any algorithm to quickly determine "zero points" (i.e. points with undefined derivation) of absolute values functions which are nested multiple times? I do know, that any part of this ...
1
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2answers
65 views

Determine the smallest number P

I have here a hard problem, which I couldn't solve. Denote $M$ the set of all functions $f:[0,1]\to\mathbb{R}$ with the following properties: $f(x)\ge0, \forall x$ in $[0,1]$, $f(1)=1$, $f(x+y)\ge ...
1
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0answers
51 views

find $x$, given $\{c_ix = k_i + y_i\}_{i=[1,n]} $

Given $$c_1x = k_1 + y_1 $$ $$c_2x = k_2 + y_2 $$ $$\vdots $$ $$c_nx = k_n + y_n $$ where the values of $\{c_1 \ldots c_n \}$ and $\{ k_1 \ldots k_n \}$ are known, and $x, \{y_1 \ldots y_n \}$ are ...
1
vote
3answers
937 views

Simplifying $\frac{a}{(a-b)(a-c)(x-a)}+\frac{b}{(b-c)(b-a)(x-b)}+\frac{c}{(c-a)(c-b)(x-c)}$

We need to simplify $$\dfrac{a}{(a-b)(a-c)(x-a)}+\dfrac{b}{(b-c)(b-a)(x-b)}+\dfrac{c}{(c-a)(c-b)(x-c)}$$ The biggest problem is that the above expression has four variables.I transformed the ...
1
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6answers
120 views

Working out the value of $a^4+b^4$

If $ab = 2$ and $a+b = 5$ then calculate the value of $a^4+b^4$ My approach: $$a^4+b^4 = (a+b)^4-4a^3b-6a^2b^2-4ab^3$$ $$=(5)^4-6(ab)^2-4ab.a^2-4ab.b^2$$ $$=(5)^4-6(24)-4ab(a^2-b^2)$$ $$=(5)^4-6(24)-...
4
votes
3answers
165 views

Simplifying $\dfrac{1}{1+x}+\dfrac{2}{1+x^2}+\dfrac{4}{1+x^4}+\dfrac{8}{1+x^8}+\dfrac{16}{x^{16}-1}$

We need to simplify $$\dfrac{1}{1+x}+\dfrac{2}{1+x^2}+\dfrac{4}{1+x^4}+\dfrac{8}{1+x^8}+\dfrac{16}{x^{16}-1}$$ The last denominator can be factored and we can get all the other denominators as ...
3
votes
1answer
90 views

Solving a system of 3 variables

How to solve or what is the algorithm to solve a system of equations like this: $$\eqalign{ (x +\phantom{3} z)^2 + (y +\phantom{3} w)^2 &= 52\cr (x + 3z)^2 + (y + 3w)^2 &= 296\cr (x -\phantom{...
1
vote
1answer
61 views

$Pr(X+Y \geq \frac{\pi}{2})$

I want to find $Pr(X+Y \geq \frac{\pi}{2})$ for joint pdf $f_{X,Y}(x,y) = x \cos y, 0 \lt x \lt \frac{\pi}{2}, 0 \lt y \lt x, 0$ otherwise. I believe I have found conditional pdf of $Y$ given $X=x$ ...
0
votes
1answer
18 views

function assistance

I think I only need a little help with this, I think I undertsand it. What I have got so far is: $12.8 = -log_{10} (H^+)$ Multiply both sides by a negative 1 (bad thing to do?) $10^{-12.8} = H^+$...
0
votes
1answer
42 views

Find $2^{2}+4^{2}….(2n)^{2}$?

I tried subtracting $1^{2}+3^{2}….n^{2}$ from $2^{2}+4^{2}….2n^{2}$, but that didn't work. I know the answer is
2
votes
4answers
367 views

Prove this identity: $\frac{2\sin^4x+\cos^2x-2\cos^4x}{3\sin^2x-1} =1$

I am stuck on this identity $$\frac{2\sin^4x+\cos^2x-2\cos^4x}{3\sin^2x-1} =1$$ I began working on the left side trying to get things to cancel out or equal one by the Pythagorean identities. I am ...
0
votes
1answer
24 views

Graphing Tangent Function

I have to graph the problem y=tan3θ-1 and then find the amplitude, period, phase shift, vertical shift, and domain and range. I have found everything except the domain. There is a vertical asymptote ...
2
votes
1answer
247 views

Find $(\delta-\alpha)(\gamma-\alpha)(\delta-\beta)(\gamma-\beta)$ as a polynomial of p,q,r,s

The equation $x^2+px+q=0$ has roots $\alpha , \beta$; the equation $y^2+ry+s=0$has roots $\delta, \gamma$. Find $$(\delta-\alpha)(\gamma-\alpha)(\delta-\beta)(\gamma-\beta)$$ as a polynomial of p,q,r,...
1
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0answers
41 views

How to simplify this radical?

How would I go about simplifying such a problem? √3(4-2√6) I'm not really used to doing this type of problem.
1
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1answer
22 views

$z=( (y⋅x)⋅0.002)+(y⋅x) $ and solve for $ x$

Sorry for such a basic question, but it has been a very long time since my high school math classes. Given the equation $z = ( (y \cdot x) \cdot 0.002) + (y \cdot x) $ how could I solve for $x$ if ...
1
vote
1answer
47 views

Factoring and Simplifying

I'm trying to do this problem, $$(4x + 1)^{15}\cdot\frac{1}{3}(12x - 5)^{-\frac{2}{3}}\cdot 12 + (12x - 5)^{\frac{1}{3}}\cdot15(4x + 1)^{14}\cdot 4$$ I've gotten down to, $$4(4x+1)^{15}(12x-5)^{-\...
0
votes
3answers
51 views

Prove the inequality $\frac{a}{b}<\frac{a+k}{b+k},(a<b, a,b,k>0)$

Help me please to prove the follow inequality: $\frac{a}{b}<\frac{a+k}{b+k},(a<b, a,b,k>0)$ thanky very much
3
votes
3answers
94 views

Algebra question: Finding inverse function

This question is about finding the inverse function of $f(x)=-\sqrt{9-x^2}$ I seem to be making an error with one of the manipulations. Here is my attempt. $$x=-\sqrt{9-y^2}$$ $$x^2=(-\sqrt{9-y^2})^...
2
votes
4answers
301 views

Solve $\cos x+8\sin x-7=0$

Solve $\cos x+8\sin x-7=0$ My attempt: \begin{align} &8\sin x=7-\cos x\\ &\implies 8\cdot \left(2\sin \frac{x}{2}\cos \frac{x}{2}\right)=7-\cos x\\ &\implies 16\sin \frac{x}{2}\cos \...
2
votes
2answers
83 views

Find the inverse of $f(x) = (x+1)/(x-8)$

Find the inverse of this function: I have gotten this far: $x = y+1/y-8$ $x(y-8) = y+1$ $x(y-8)-1=y$ $xy-8x - 1 = y$ I think I went backwards?
0
votes
1answer
65 views

Why does $\dfrac{2}{\sqrt{-4x^2-4x}}$ simplify into $\dfrac{1}{\sqrt{-x^2-x}}$?

Why does $\dfrac{2}{\sqrt{-4x^2-4x}}$ simplify into $\dfrac{1}{\sqrt{-x^2-x}}$? What's going on here? How is it being simplified?
0
votes
1answer
67 views

Polynomial Problem $P(x)=x^{100}-100x+99$

Let $P(x)=x^{100}-100x+99$. Prove: i) $x=1$ is a double root ii)The polynomial has no other real roots Thanks in advance!
2
votes
2answers
74 views

Volume of a sphere section. Is there enough data?

I am thinking about a problem that seems easy but I suspect that is not possible solve it with the available data. Thanks very much! Problem is this: There is a sphere cutted in four equal ...
4
votes
3answers
150 views

Game With 21 Squares, How Many Possible Answers? Function Building

We played this game in our math class, okay, I'll explain how it's played. There are 21 squares in a straight line across, the first person shades in 2 adjacent squares. The next player shades in 2 ...
3
votes
5answers
102 views

Definition of square root symbol - $\sqrt{x^2}=?$

If ${f(x) = \sqrt{x^2}}$, then f(x) can also be expressed as: C. ${|x|}$ D. $ \pm x$ I thought the answer was D, but it's C. Couldn't it be both?
2
votes
3answers
449 views

How to calculate this shape's volume

So I've got this shape How would I calculate the volume? I thought about splitting it up into a cone somehow but I don't have the rest of the information to do that, I think...What's to do? This ...
0
votes
1answer
51 views

Pre-Calculus definition Clarification

The x-axis is a horizontal asymptote for the exponential function $f(x) = a^x$. This is because when $a>1$, we have $a^x \to 0$ as $x \to -\infty$, and when $0 < a < 1$, we have $a^x \to 0$ ...
1
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0answers
74 views

Arithmetic and Algebra exercises on latex source code.

I´m currently writing a little book for two student that I teach. The book covers school arithmetics and algebra, and it include theory and examples. Since I don´t have time to prepare a good sets of ...
1
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1answer
57 views

Determine all real solutions of the system of n equations

For $n\geq3$, determine all real solutions of the system of $n$ equations: $x_{1}+x_{2}+...+x_{n-1}=\frac{1}{x_{n}}$ ... $x_{1}+x_{2}+...+x_{i-1}+x_{i+1}+...x_{n}=\frac{1}{...
1
vote
2answers
107 views

Is $-1=\sqrt{1}$ true?

I need to find the solutions of $$x=\sqrt{5+4x}.$$ I found that $x=5 \vee x=-1$. But is $-1=\sqrt{1}$ really true?
0
votes
3answers
60 views

Solve $ | 2x+4 | > |x-1| $

I know there are many ways to solve it, but i want some help to understand why it is not working specifically on this way: a)For $\,\,x<-2$, the inequation becomes: $-2x-4>-x+1 \...
1
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1answer
62 views

$P=\left\{\theta:\sin \theta-\cos \theta = \sqrt{2}\cos \theta\right\}$

Let $P=\left\{\theta:\sin \theta-\cos \theta = \sqrt{2}\cos \theta\right\}$ and $Q=\left\{\theta:\sin \theta+\cos \theta=\sqrt{2}\sin \theta\right\}$ be two sets , Then which one is Right. $(a)\;\;\;...
2
votes
2answers
52 views

Proving a function is injective using the definition

The definition of an injective function is $f(x_1)=f(x_2) \implies x_1 = x_2$. I am having trouble understanding at what point into the proof do you give up and conclude that a function is not ...
1
vote
2answers
44 views

Proving a function is one to one over a domain and codomain

I know that the definition of a one-to-one function is $f(x_{1})=f(x_2) \implies x_1 = x_2$. I am having trouble understanding how to prove that a function is one-to-one. This was a given example: $f(...
1
vote
1answer
49 views

Two functions intersect,solve the equation.

Given two functions,show where they intersect $(x^2−5)^2/(x+7)^2=\sqrt{169-x^2}$ I have already tried to square both of them but I get a very complex equation and I can not solve it. I saw a guy who ...
0
votes
1answer
32 views

Polynomial multiplication

If a function Q(x) only has integer coefficients and has factors : f1(x),f2(x),... fn(x). Can the functions f1(x),f2(x),... and fn(x) necessarily have integer coefficients? Gaussian integers are fine ...
2
votes
2answers
54 views

Proving identites with only tan and cot

I can't figure out how to start this identity or solve it. I began on the right side and distributed but I don't know where to go from there. cotA + cotB= cotAcotB (tanB + tan A)
1
vote
2answers
45 views

Exponents in Identities

I am stuck on this identity $\sin^3 x \cos x-\sin^5 x \cos x= \sin^3 x \cos^3 x$ I began working on the left side, but can't seem to reduce the exponents
0
votes
1answer
112 views

Solving Triangle Word Problems

A ship leaves port for a remote island 34 miles west and 60 miles south of the port. Find the distance and bearing to the remote island. What would the diagram look like to this problem? and what ...
1
vote
0answers
27 views

MultiEquations (with fractions)

Can you please help me solve these equations i don't understand how to solve them with fractions. 1=n-2/15 151/20 =2a+1 3/4 -3/5 -2 1/5k = - 26/25
0
votes
2answers
31 views

Possibility of integral quadratic with these roots

If x and w are the roots of a quadratic equation with integral coefficients then is this possible: ${x = w = \frac{2}{3}}$. The correct answer says it is, but how is that so if it means: ${(x-\frac{2}{...
4
votes
4answers
211 views

Can $x^3+x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve it: $x^3 + x^2 + 1 = 0$ Am I missing something here, or is indeed a more advanced method necessary to solve ...