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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0
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4answers
55 views

Simplify the difference quotient

Simplify the difference quotient $$\frac{f(x)-f(a)}{x-a}, $$ by rationalizing the numerator, where $f(x) = \dfrac{-3}{\sqrt{x}} $. Please help, I am stuck. The textbook says the answer is: ...
2
votes
2answers
119 views

Find a positive number $\delta<2$ such that $|x−2| < \delta \implies |x^2−4| < 1$

I have to find a positive number $\delta<2$ such that $|x−2| < \delta \implies |x^2−4| < 1$. I know that $ \delta =\frac{1}{|x+2|} $ has this behaviour, but it is not guaranteed for it to ...
2
votes
3answers
56 views

Roots of Unity - $x^3 = -i$

I need to find the roots of unity for the following equation: $$x^3 + i = 0$$ Thus, $x^3 = -i$. I know that $-i = \exp[i(\frac{3\pi}{2} + 2n \pi)]$ however I do not know how to get all roots. ...
2
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4answers
93 views

Dividing units with exponents

I am having troubles learning how to divide units that have negative exponents. I have tried multiple times and I am not sure if i am doing it right. This is the question: ...
0
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2answers
59 views

Usage of implies and equivalence in mathematics, confusion?

I am confused because I have seen implies and equivalent used interchangibly. For instance, I've seen $$x-y=0 \implies x=y$$ And I've also seen $$x-y=0 \Longleftrightarrow x=y$$ Are both of ...
0
votes
1answer
125 views

How to expand this polynomial division?

My Physics teacher gave me a problem and its solution, what I have todo is to expand the solution, but when I do it I do not get to the same solution he says is the right one, here is the problem: ...
2
votes
1answer
40 views

Recurrence relation of the following sequence?

This is the code: for (unsigned int i = 0; i < n; ++i) if (i % 2 == 0) ++k; And this is the output for when ...
0
votes
2answers
193 views

Factor $ x^3-3x^2-4x+12$

How do I go about factoring this problem? What is the best method? I can not factor out an $x$ since the $12$ does not have a variable. I usually use the Criss Cross method.
4
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3answers
4k views

Find all values for cos(i)

In my Differential Equations class recently we have learned about Euler's Formula and Fourier Series. I am given the problem ...
0
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1answer
46 views

Polynomial Long Division in Algebra

How do i even begin to fathom these questions? How do i begin to answer them? Help would be much appreciated! Divide $X^5 - X^4 - 6X^3 - 8X^2 + 8X +48$ , by $X^2 - X - 6$. Hence fully factorise $X^5 ...
0
votes
2answers
29 views

Compositions with Restricted Domain.

Hey Guys! How do you do this problem. There is no overlapping domain for f(x) and g(f(x)). Detailed explanation as to how to do the problem will be preferred! Thanks.
9
votes
1answer
97 views

Solving $x_1+x_2=x_3^2, x_2+x_3=x_4^2, x_3+x_4=x_5^2,x_4+x_5=x_1^2, x_5+x_1=x_2^2$ in reals

find answers of this system of equations in real numbers$$ \left\{ \begin{array}{c} x_1+x_2=x_3^2 \\ x_2+x_3=x_4^2 \\ x_3+x_4=x_5^2 \\ x_4+x_5=x_1^2 \\ x_5+x_1=x_2^2 \end{array} \right. ...
1
vote
2answers
166 views

Newton 2nd law for rotations: $\tau = I \alpha$ dimensions correct?

I have a really stupid question, which I can't figure out at the moment. I don't see why the following is correct when you check the dimensions: $\tau = I \alpha \;\;\rightarrow \mathrm{Nm= kg ...
1
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2answers
99 views

Turning a decimal back into it's Reciprocal?

So when you have a number: 5, the reciprocal would be 1/5 and in decimals 0.2 To go from decimal back to the original number i can just go: 0.2 x 5^2 = 5 But what happens when i don't know the ...
1
vote
5answers
127 views

Not clear on what we mean with numbers with infinite digits

I am confused on a rather simplistic question. 1/3 = 0.333333333333 to infinity. So it has infinite digits. How is it possible to multiply such a number with another one and get a finite number? 6/3 = ...
-1
votes
1answer
38 views

Stuck with factoring an expression

$$(x^{ 2 }+2)^{ 5/2 }+2x(x^{ 2 }+2)^{ 3/2 }+x^{ 2 }\sqrt { x^{ 2 }+2 } $$ I believe that the first step to factoring this would require me to take the factor with the smallest exponent out of the ...
0
votes
1answer
49 views

Show that $x^n = a$ has at most one real positive real if $n$ is a positive integer

Show that $x^n = a$ has at most one real positive real if $n$ is a positive integer. I can solve this question by drawing graphs for different $n$. But how should I approach the problem if I want to ...
0
votes
1answer
158 views

Graphing inverse of parabola, should the domain be restricted or not?

Find the inverse of $f(x)=y=x^2-2x+6$ for $x \ge 1$ The inverse of that above function is $|y-1|=\sqrt{(x-5)}$ The domain of this function requires $x>5$. There are now two functions to graph, ...
1
vote
4answers
55 views

Third point of a triangle in the complex plane

I have an equilateral triangle with two points equal to $(2+2i)$ and $(5+i)$. I want to find the third point(s) (there are $2$ of these). I have that the side length of the triangle is $\sqrt{10}$.
1
vote
1answer
94 views

Compositions and Restricted Domains

Hey guys. I got the answer under question 3 (which is circled). Can you please verify if it is correct? If not, can you please specify how to go about this problem? Thanks
-2
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1answer
73 views

Complex numbers and geometry

There exist two different complex numbers $c_1$ and $c_2$, that together with $2+2i, 5+i$ form the vertices of two equilateral triangles. Find the product $c_1c_2$.
0
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1answer
105 views

Complex Number Geometry 5 [closed]

Let S be the set of complex numbers z such that the real part of 1/z is equal to 1/6. This set forms a curve. Find the area of the region inside the curve. Can someone explain this problem to me?
1
vote
1answer
8 views

Not understanding the answer to a fractional expression

$$\frac { 8r^{ 1/2 }s^{ -3 } }{ 2r^{ -2 }s^{ 4 } } $$ The first step I took was getting rid of the negative exponents: $$\frac { 8r^{ 1/2 }2r^{ 2 } }{ s^{ 3 }s^{ 4 } } $$ Then I performed the ...
2
votes
2answers
61 views

Showing if $n \ge 2c\log(c)$ then $n\ge c\log(n)$

Is this true that if $n \ge 2c\log(c)$ then $n\ge c\log(n)$, for any constant $c>0$? Here $n$ is a positive integer.
0
votes
1answer
283 views

Distance, Speed and Time Word Problem

The word problem: Mark walks 2000 feet west and 600 feet north of his starting position. In the side walk the speed is 6 ft/sec and 4 ft/sec through the grass. How far should he walk on the sidewalk ...
0
votes
2answers
82 views

How to find the composition of two functions and its domain?

I have no clue how to go about this problem. A detailed explanation would be preferred. Thanks
0
votes
1answer
85 views

How to find the domains of functions $f(x) = x-5$, $g(x) = \sqrt{x-5}$, and of their sim?

I've been studying on Study Plan Practice, on MyMathLab for my College Algebra class. We're going over the Algebra of Functions right now and several things don't make much sense. The question is: ...
0
votes
1answer
145 views

How do I evaluate the summation of a maximum function?

Question: $f(x)=max_{a∈[1,−1]} \sum^d_{j=1}ax_j$ and $g(x)=\sum^d_{j=1}max_{a∈[1,−1]}ax_j$. and where $x=(x_1,…,x_d)∈\Bbb R ^d$ is a real vector. What is the relationship between $f(x)$ and $g(x)$? ...
0
votes
2answers
37 views

How would I find the second asymptote of the following function:

How would I find the second vertical asymptote of $(2x^2)/(6x^2+11x-10)$? I know that the first one is 2.5 from looking at a graphing calculator, but the second one is a small decimal asymptote, ...
0
votes
1answer
71 views

Diophantine equation: $2(x^3+xy+y^3)=3(x+y)$

Here is a nice equation: $2(x^3+xy+y^3)=3(x+y)$ over $ \mathbb{Z}$ x $\mathbb{Z}$. Any nice way to approach this?
2
votes
3answers
378 views

Show that two expressions are equivalent

I am trying to prove a hyperbolic trigonometric identity and I ran into the following expression: $$\frac{\left (\sqrt{x^2+1}+x \right )^2+1}{2\left ( \sqrt{x^2+1} + x \right )} \quad.$$ This ...
0
votes
1answer
86 views

How is 1 cubic decimeter =1 liter; and 1000 cubic centimeters equal to 1 liter?

Am I cubing these units? If so how? My thinking? decimeter =10^-1 = 0.1, centimeter 10^-2= 0.01 If 1 liter = 1 cubic decimeter, how can it also equal 1000 cubic centimeters when the two units are ...
0
votes
6answers
354 views

How to solve a system of two linear equations with two unknowns?

How do I solve this system of equations? $$\begin{cases} 7(a+b)=b-a \\4(3a+2b)=b-8\end{cases}$$ Progress I tried both substitution and elimination, but when I set $a$ or $b$ free on one side, I ...
1
vote
4answers
90 views

Seeking verification: $\sqrt[\large 3]{a \cdot \sqrt a} = \sqrt a\quad?$

$\sqrt[\large 3]{a \cdot \sqrt{a}}=?$ Is the answer simply $\sqrt{a}\quad?$
10
votes
5answers
160 views

Prove $3(\sin x-\cos x)^4 + 6(\sin x+ \cos x)^2 + 4(\sin^6 x + \cos^6 x) -13 = 0$

Q) Prove that $3(\sin \theta-\cos \theta)^4 + 6(\sin \theta+ \cos \theta)^2 + 4(\sin^6 \theta + \cos^6 \theta) -13 = 0$ Source: Trigonometric Functions, Page 5.9, Mathematics XI - R.D. Sharma ...
1
vote
1answer
84 views

Solve $\sqrt x = x/2$

If $f(x) = \sqrt x$ and $g(x) = x/2$. Find the area of this limited area between $f(x)$ and $g(x)$. I'm having trouble to solve this equation $\sqrt x = x/2$ that should give me the x values. I know ...
2
votes
1answer
66 views

Solve $x+\frac{2}{y}=3,y+\frac{2}{z}=3,z+\frac{2}{x}=3 $ in reals

Find answers of this system of equations in real numbers$$ \left\{ \begin{array}{c} x+\frac{2}{y}=3 \\ y+\frac{2}{z}=3 \\ z+\frac{2}{x}=3 \end{array} \right. $$ Things I have done: first I ...
4
votes
1answer
84 views

Show that the equation $a_1e^{\alpha_1x} + a_2e^{\alpha_2x} + \cdots + a_ne^{\alpha_nx} = 0$ has at most $n - 1$ real roots.

For non-zero $a_1, a_2, \ldots , a_n$ and for $\alpha_1, \alpha_2, \ldots , \alpha_n$ such that $\alpha_i \neq \alpha_j$ for $i \neq j$, show that the equation $$a_1e^{\alpha_1x} + a_2e^{\alpha_2x} + ...
1
vote
4answers
280 views

Are these proofs logically equivalent?

Here are two proofs, firstly: x = 0.999... 10x = 9.999... = 9 + 0.999... = 9 + x 9x = 9 x = 1 And secondly: ...
0
votes
1answer
3k views

Aquiring Triangular Signal Equation from Waveform

I've looked everywhere and even the textbook does not explain how to do this. This is probably very simple, yet I can't figure it out. How do you derive the expression at the bottom for the ...
3
votes
3answers
79 views

Prove $\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0$ if $\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0$

if $a,b,c$ are real numbers and $$\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0$$ Prove $$\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0$$ things i have done: using the assumption i ...
0
votes
1answer
30 views

Maximal Domain for the following functions

Hi I am quite lost with understanding maximal domains. I understand that the numerator when dealing with maximal domains usually are not taken into consideration. Is this true for all cases? Can ...
2
votes
3answers
65 views

Inequality: $2(p^2+q^2+r^2)+2(pq+qr+rp)\ge pqr$

I need to determine the range of $p,q,r$ such that $2(p^2+q^2+r^2)+2(pq+qr+rp)\ge pqr$. I am not given any other information except that $p,q,r\in \mathbb{R}$. I haven't solved a problem like this ...
1
vote
1answer
34 views

An inequality related to Pythagorean theorem: if $A^{2} + B^{2} = C^{2}$, then $A+B>C$

If $A^{2} + B^{2} = C^{2}$, prove $A+B>C$ for all $A>0$ and $B>0$ Intuitively it seems to apply to all positive real numbers(since the hypotenuse of a right triangle is shorter than the sum ...
2
votes
8answers
209 views

Inequality: $x^2+y^2+xy\ge 0$

I want to prove that $x^2+y^2+xy\ge 0$ for all $x,y\in \mathbb{R}$. My "proof": Suppose wlog that $x\ge y$, so $x^2\cdot x\ge x^2\cdot y\ge y^2\cdot y=y^3$ (because $x^2\ge 0$ so we can multiply ...
0
votes
2answers
64 views

Determine the number of digits in $4^n$

Let $n$ be a natural number. How can we determine the number of digits in $4^n$? For example $4^{20}$ has $13$ digits.
3
votes
2answers
31 views

Require help with Inequality problems

I am unable to find the solution for below Inequality problems. 1) $2/x<3$ The answer seems to be x belong to $(-\infty,0)\cup (2/3,\infty)$ 2) $\dfrac{x+4}{x-3}<2$ The answer seems to be x ...
1
vote
0answers
30 views

Integer solutions to a trig equation. Irrational numbers etc.

Can the equation $$3\left(1\pm 2\cos\left(\frac{2\pi x}{n}\right)\right)=\left(1\pm 2\cos\left(\frac{2\pi y}{n}\right)\right)\left(1\pm 2\cos\left(\frac{2\pi z}{n}\right)\right)$$ Ever have a ...
0
votes
1answer
89 views

$f(x+h)$ not equal to $f(x) +f(h)$???

I'm taking College Algebra at a local community college, and I just wasn't able to follow how my professor came to these conclusions. (3 separate times.) $$\frac{f(x+h) - f(x)}{h},$$ $$f(x) = ...
0
votes
1answer
136 views

Proof that if an algebraic integer is rational, it is integer?

There is a well-known fact that the intersection of 𝔸 and ℚ is ℤ. It is mentioned in many places, including Wikipedia, without proof. Does this theorem have a well-known name, and where can i find ...