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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1answer
35 views

How do I create a function from this code? [closed]

Here is the code: for (int i = 1; i < n; i *= 2) ++k; I need to express this as a function. I don't know where to begin.
0
votes
0answers
41 views

Simplifying the sum of two rational functions

The original equation is: $$\frac{4x}{4x+1} + \frac{3}{16 x^2-1}$$ I am supposed to factorize then simplify it. Now after I factorized it I got $$\frac{4x(4x-1) + 3}{(4x+1)(4x-1)}$$ but I couldn't ...
0
votes
1answer
17 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
4 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
53 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
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1answer
16 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
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2answers
40 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
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1answer
54 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
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1answer
18 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
3answers
52 views

I need to solve for $x$, where do I start? [closed]

I need to solve for $x$. $y$ is $5$ in all cases. $$ y = 1.5 \frac{(3x - 2)}{6} $$
0
votes
2answers
16 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
62 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?
0
votes
0answers
18 views

Find the running time of the following program fragment

The exercise in my book is asking me to calculate the running time of the following for loop: for (int i = 0; i < n; ++i) ++k; This instantly reminds me ...
2
votes
3answers
121 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
26 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
298 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 ...
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4answers
75 views

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

$\sqrt[\large 3]{a \cdot \sqrt{a}}=?$ Is the answer simply $\sqrt{a}\quad?$
8
votes
5answers
117 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 ...
0
votes
1answer
57 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 ...
0
votes
1answer
191 views

Evaluate sum of floor function [closed]

How to find the sum : N is less than 10^10 This is a problem from a live contest at CodeChef - http://www.codechef.com/SEPT14/problems/FLOORI4/
0
votes
1answer
43 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 ...
3
votes
1answer
74 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
votes
1answer
37 views

Small quiz questions [closed]

I was lately doing a quiz when I came across these question. 1)A car has two small holes. One, by itself, would make the tire flat in 9 minutes and the other in 6. How long will it take both holes to ...
2
votes
4answers
202 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
10 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
63 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
16 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
56 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
30 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 ...
0
votes
8answers
123 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
53 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.
2
votes
2answers
28 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
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0answers
19 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
76 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
44 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 ...
1
vote
1answer
35 views

How many points to span a goniometric wave and how to construct the goniometric function

I have two questions concerning the spanning of a simple trigonometric function: What is the minimum number of points to define/span a "simple" trigonometric wave in two dimensions? Is it possible ...
2
votes
4answers
79 views

Showing Surjectivitity of $f(x) = x^3$

I want to show that the function $f: \mathbb{R} \to \mathbb{R},\; f(x) = x^3$ is surjective. First Question: If a function has an inverse, it is bijective yes? Second Question: Is my process ...
1
vote
3answers
38 views

Solve $ x^2+y^2=4, z^2+t^2=9, xt+yz=6 $ in integers

find answers of this system of equations in integers$$ \left\{ \begin{array}{c} x^2+y^2=4 \\ z^2+t^2=9 \\ xt+yz=6 \end{array} \right. $$ things I have done: we can observe that ...
2
votes
2answers
31 views

Prove $a^4+b^4+(a-b)^4=c^4+d^4+(c-d)^4$ if $a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$

if $a,b,c,d$ are positive real numbers and $$a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$$ Prove $a^4+b^4+(a-b)^4=c^4+d^4+(c-d)^4$ Things i have done: from assumption $a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$ I ...
8
votes
5answers
1k views

Why add before dividing in this equation?

For the following equation, I know the correct answer is $9$: $$ x / 3 + 2 = 5 $$ You subtract $2$ from each side, and the multiply each side by $3$... But why do you subtract the $2$ first? Doesn't ...
1
vote
1answer
27 views

Solving $\frac{\log(x)+c_1}{x}=c_2$, where $c_2 < e^{-1}$

There is an answer when $c_1=0$ at (Solving $\frac{\log(x)}{x}=c$, where $c < e^{-1}$). How could we solve the following? $$\frac{\log(x)+c_1}{x}=c_2, \text{where } x > 1,~c_1>0, ...
0
votes
1answer
27 views

Why does the sign change here?

They give the recurrence relation as: $$T(n) − 4T(n − 1) + 3T(n − 2) = 0,\ T(0) = 0,\ T(1) = 2$$ And then they say it can be written as the following for $n > 1$: $$T(n) = 4T(n − 1) − 3T(n − 2)\ ...
1
vote
2answers
27 views

should rate of change be negative

Say I have a spherical snowball. I want it's average rate of change of surface area as radius goes from 25cm to 20cm. I did the calculation. $f(r)=4*\pi*r^2$. That's the formula of surface area of ...
1
vote
2answers
33 views

Help with proof by induction

The author generates a Tower of Hanoi and looks at the sequence: $$1, 3, 7, 15, 31, 63,...$$ He guesses the recurrence relation from the first few terms: $$H_{n} = 2^{n} - 1$$ Now he wants to ...
1
vote
0answers
57 views

Solving ${c_1}^x+\sqrt{\frac{\log(x)x}{2}}+3\log(x)x \le c_2$

Is there any way to solve $${c_1}^x+\sqrt{\frac{\log(x)x}{2}}+3\log(x)x\le c_2,$$ for $x>1$, $0<c_1<1$, and $0<c_2<<1$? Thanks
1
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4answers
26 views

Showing an Absolute Value Inequality Problem Proof

I tried solving this question but it does not works for me. Q.) Show that $\left|x + \frac1{x}\right| \ge 2$ for all $x \ne 0$ There are two ways to do. One is squaring and other is to use absolute ...
3
votes
1answer
45 views

Solving $\frac{\log(x)}{x}=c$, where $c < e^{-1}$

I am just wondering if there is an easy way to solve $$\frac{\log(x)}{x}=c, \text{where } x > 1 \text{ and } c < e^{-1}$$
1
vote
3answers
62 views

Way to evaluate this algebraic expression by hand without going insane?

EDIT: fixed asymmetrical denominator! I was following along with a proof of Routh's theorem, and the final expression for the area of the enclosed triangle is $$ 1 - \frac{x}{xz + x + 1} - ...
-1
votes
2answers
52 views

Find the other 2 points of a rectangle? [closed]

$PQRS$ is a rectangle with vertices $P(-4,-1)$ and $Q(-6,5),$ and $PQ=2(QR).$ Find the coordinates of $R$ and $S$? I'm so stuck please help! There are 2 answers for each point. almagest has the right ...
1
vote
3answers
63 views

Basic Algebra Inequality Proof

Is there a formal proof that $x<2 \iff -x>-2$, or is this just a matter of convention?