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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4
votes
0answers
46 views

Rationality and triangles

Consider a triangle with angles $\alpha, 5\alpha, 180-6\alpha$. What is the minimum perimeter of that triangle, if it has integer sides and $5\alpha<90$?. Let's call tha sides that face each ...
15
votes
7answers
960 views

How to make a “function”?

I dropped out of school early when I was still a teenager and now I'm trying to take my GED. I'm really close to passing but I'm still having trouble understanding some concepts. In the pre-test, ...
1
vote
5answers
300 views

How to prove that $\frac{x}{a} + \frac{y}{b} = 1$ where $a$ is $x$-intercept and $b$ is $y$-intercept

How to prove that $\dfrac{x}{a} + \dfrac{y}{b} = 1\;$ where $\,a\,$ is the $\,x$-intercept and $\,b\,$ is the $\,y$-intercept for all $\,a,b \neq 0$ This was a question on my son's math analysis ...
1
vote
1answer
20 views

Maximum and minimum of a fractional function

Let $x, y \in \mathbb{R}$, $a, b, c$ are three real parameters with $c\neq 0$. Find the maximum and minimum of $\dfrac{ax+by+c}{\sqrt{x^2+y^2+1}}$ This is quite complicated if I calculate the ...
0
votes
0answers
16 views

Existence of solution for an equation including polynomial and trinogometric sum

Prove that the following equation has at least a solution in $[-\pi, \pi]$ : $$ x^5+\sum^{n}_{k=1}(a_k\cos kx+b_k\sin kx)=0 $$ I think the existence of the solution on $[-\pi, \pi]$ strongly depends ...
3
votes
1answer
74 views

$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational

If $m_1 , m_2, \cdots m_n$ are natural numbers where at least one of them is not a perfect square, then how do I prove that the sum $$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$$ is irrational? I'm ...
0
votes
7answers
653 views
3
votes
1answer
304 views

Why is $ab+bc+ac = 0$ in some situation?

This is originally a Computer Science question, but I ran a equation that is too hard to solve. Here goes. So the problem is quite simple, given positive integers $a$, $b$, $c$, and calculate ...
0
votes
2answers
31 views

variables to the power of a fraction

I have this question for advanced math, I can't seem to get my head around. $$\frac{x^{5/2}}{(x^{1/3})^4}$$
2
votes
3answers
52 views

$x$-intercept of cosine graph

I am having problems understanding how to find the $x$-intercept of a cosine graph. Example: $10\cos(x/2)$ Answer:$((2n + 1)\pi , 0 )$ I have the answer just need help understanding the steps, ...
0
votes
1answer
15 views

Exclude one function from another

Is it possible to find a function, $g(n)$ that will include all the natural values except those in $f(n)$? $$f(n) = 3n$$ $$g(n) = 1,2,4,5,7,8,10...$$
0
votes
1answer
38 views

How to solve quartic polynomial equation

Can someone tell me how to solve $x^4 + 6x^2 + 5 = 0$? I know what to do when each term has an exponent one less than the previous term (e.g., $x^4 + 3x^3 + 6x^2 + 5 = 0$), but not when exponents are ...
0
votes
0answers
10 views

Displaying a 3D function without a graph

I have a 3D function $z=\dfrac{x}{y}$, and I have no access to a function grapher, but I still need to display this function in a comprehensible way. I thought of a table, but even with a domain of ...
-1
votes
0answers
17 views

Combing piecewise functions [on hold]

How would I combine the following two piecewise functions in terms of addition and subtraction? How would I find $f(x) + g(x)$, and also $f(x) - g(x)$? Thanks!
1
vote
4answers
79 views

Using mathematical induction to show that for any $n\ge$ 2 then $\prod_{i=2}^n\bigl(1-\frac{1}{i^2}\bigr)=\binom{n+1}{2 \cdot n}$

I'm trying to work through some practice problems but I've been stuck on this for god knows how long now and I've no idea where to even start. Just wondering if it would be possible for someone to ...
0
votes
0answers
12 views

Iteratively solve this equation

I am supposed to solve $1 = \left( \frac{\mu}{f} \right)^{\frac{3}{2}} \left( 1+ \frac{ \pi^2}{8} \left( \frac{kT}{\mu} \right)^2 \right)$ iteratively for $\mu$ and am supposed to get $$\mu = f ...
0
votes
1answer
64 views

sum of exponentials to non-integer power

I have the expression \begin{equation} (e^{at}+e^{bt}+e^{ct})^{v} \end{equation} for some a,b,c which isn't important. I'd like to take a limit $t\rightarrow \infty,v\rightarrow 0,vt=\text{constant}$. ...
0
votes
2answers
23 views

Line Intersect in Diagonals of a Rectangle

The diagonals of the rectangle have these equations: $$y = 4x-10\\ \\ y = -4x+18$$ Find the point at which the diagonals intersect. First, I tried working out $(x,y)$ $4x - 10 = -4x + 18$ $4x = ...
-1
votes
0answers
29 views

Slope of the secant line given a point

The point $P(1,0)$ lies on the curve $y= \sin(10\pi/x)$. (a) If $Q$ is the point $(x, \sin(10\pi/x))$, find the slope of the secant line $PQ$ (correct to four decimal places) for $x=2, 1.5, 1.4, 1.3, ...
2
votes
1answer
60 views

Closed form for the summation $\sum_{k=1}^n\frac{1}{r^{k^2}}$

Is there any closed form for the finite sum $$\sum_{k=1}^n\dfrac{1}{r^{k^2}}$$ or infinite sum ( when $|r|<1$) $$\sum_{k=1}^\infty\dfrac{1}{r^{k^2}} ?$$ While solving this problem, I found this ...
3
votes
4answers
477 views

Given that a,b,c are distinct positive real numbers, prove that (a + b +c)( 1/a + 1/b + 1/c)>9

Given that $a,b,c$ are distinct positive real numbers, prove that $(a + b +c)\big( \frac1{a}+ \frac1{b} + \frac1{c}\big)>9$ This is how I tried doing it: Let $p= a + b + c,$ and $q=\frac1{a}+ ...
28
votes
10answers
4k views

Explanation of method for showing that 0 / 0 is undefined

(This was asked due to the comments and downvotes on this Stackoverflow answer. I am not that good at maths, so was wondering if I had made any basic mistakes) Ignoring limits, I would like to know ...
0
votes
0answers
8 views

Bound all $k$-th derivatives by directional derivatives of order $k$

Assume $f\in C^k(\mathbb{R}^n)$, $x\in\mathbb{R}^n$, and $|(\partial_\xi)^kf(x)|\leq 1$ for all $\|\xi\|=1$. Which bounds do we have for $|\partial^\alpha f(x)|$ when $|\alpha|=k$? For example, if ...
31
votes
2answers
942 views

Intuitive reasoning why are quintics unsolvable

I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, ...
1
vote
0answers
73 views

Crossed Ladders Problem

Two ladders, one 10 meters long and the other 8 meters [long], have been placed in a trench as indicated in the opposite figure. Their point of intersection, M, is 3 meters from the base of the ...
0
votes
2answers
268 views

Linear Programming Problem?

You are about to take a test that contains computation problems worth 6 points each and word problems worth 10 points each. You can do a computation problem in 2 minutes and a word problem in 4 ...
0
votes
2answers
28 views

General Formula for Principle Square Root of Complex Number

How can I prove that $ \sqrt{z} = \sqrt{|z|} \frac{(z + |z|)}{|z+|z||} $ without using mathematical induction, and if I cannot -- how would I go about using induction in the set of complex numbers ?
3
votes
1answer
49 views

Solving A Certain Diophantine Equation

I am stack on finding the solution of the diophantine equation: $d(2^{k+1}-1)-b^2(2^{k+1}-2)=1$. where $k\geq 1$ and $b^2>d$ for $b$ an odd composite integer. Is there a solution to this ...
-1
votes
0answers
17 views

Which of the Following statements are true? algebra 2 [on hold]

I need help with this problem. Help me find out which one of the statements are true. There can be more than one. I'm positive that one of them is A.
0
votes
2answers
36 views

Forming equations for exponential growth/decay questions

Problem Dry cleaners use a cleaning fluid that is purified by evaporation and condensation after each cleaning cycle. Every time the fluid is purified, 2.1% of it is lost. The fluid has to be topped ...
0
votes
1answer
687 views

How to combine an amount of money with the compound interest function?

Tommy has some money at home from his graduation modeled by the function $h(x)=350$. He read about a bank that has savings accounts that accrue interest according to the function $s(x)= 1.04 ...
10
votes
3answers
9k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...
0
votes
0answers
28 views

Logarithms: expanding, condensing, inverse, and checking for extraneous solutions. [on hold]

They're all separate so one answer doesn't apply to others, but I need help with how to condense logarithms, find the inverse, and check for extraneous solutions. First I'm condensing logarithms, ...
7
votes
12answers
1k views
+50

Irrational numbers in reality

I have a square stone slab 1 metre by 1 metre, by the Pythagorean identity the diagonal from one corner to another is given by $\sqrt 2$. However $\sqrt 2$ is an irrational number, could someone ...
0
votes
2answers
35 views

Show$\:\frac{1}{\left|x^2+x+1\right|}\:\ge \:\frac{1}{x^2-\left|x\right|-1}$

This is the answer I can come up with. I get the complete opposite of what I'm supposed to get. My mistake is probably in the first part, could anyone help me out? $$\left|x^2+x+1\right|\:\ge ...
-5
votes
3answers
40 views

The lines with equations $y = 5x − 6$ and $10x + cy = 8$ are perpendicular, find c [on hold]

The lines with equations $y = 5x − 6$ and $10x + cy = 8$ are perpendicular. Find the value of c. Well, I am not sure even where to start
0
votes
1answer
49 views

$(a+\frac{1}{2})^n + (b+ \frac{1}{2})^n$ is an integer for at most finitely many $n$ [duplicate]

Prove that for any positive integers $a,b$ $(a+\frac{1}{2})^n + (b+ \frac{1}{2})^n$ is an integer for at most finite number of integers $n$. Here is what I tried ; I tried to use mathematical ...
-4
votes
1answer
35 views

Algebra,complex numbers home work problem [on hold]

Please I want the solution of this problem : $z= \dfrac{(2-i) \cdot (x+4i)}{3-4i}$ and $|z|=2$ then $X=?$
0
votes
5answers
49 views

Number theory proof [on hold]

$(i)$ Prove that for every natural number $n \geq 2$, one has $(n + 1)|(n^3 + 1)$; $(ii)$ Suppose that $n$ is a natural number exceeding $1$. Prove that $(n^2-1)|(n^3+1)$ if and only if $n = 2$.
2
votes
7answers
638 views

Working Out Easy Equations

does anyone know how to do this equation? I know it's easy but I can't work out what the question means. When I expanded the first equation: $(y+4)-(y-3)$ $y^2 -3y +4y - 12$ $y^2-1y-12$ Not ...
1
vote
3answers
5k views

Given $f(x)$ its inverse function, domain and range

$f(x) = \frac{{2x + 3}}{{x - 1}},\left[ {x \in {R},x > 1} \right]$ I've got the inverse function to be: ${f^{ - 1}}(x) = \frac{{x + 3}}{{x - 2}}$ How would I go about working out the range and ...
3
votes
3answers
2k views

Useful trigonometry tricks/shortcuts

I'm curious as to any "tricks" or shortcuts that could help make verifying/solving trigonometric identities easier, for example one is: $$a\cos\theta+b\sin\theta = \sqrt{a^2+b^2}\,\cos(\theta-\phi)$$ ...
0
votes
1answer
52 views

what is going on here?

Suppose we have a function $f(x), D:( -\infty,0)\cup (0,\infty)$ and for which $$f'(x) = \frac{x^3-1}{x^3} $$ Apparently there is only one point of extremum here, $x=1$, however upon reviewing the ...
12
votes
2answers
138 views

Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$

Find all functions $f$ such that $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$ Let us plug in $n=1$ $f(f(1))+f(2)=3$ Since the function is from $\mathbb{N}$ to $\mathbb{N}$, ...
0
votes
1answer
47 views

Polynomials, prove exercise question about question

There is a polynomial P with integer coefficients and with pairwise different integers $a,b,c$ . Prove that it is not possible for $P(a) = b$, $P(b)=c$, $P(c) = a$ First off I don't understand ...
1
vote
1answer
24 views

Finding an Expression for the Difference of Roots of the Quadratic Equation

Let the equation $ax^2+bx+c=0$ have the roots $\alpha$ and $\beta$, then what is $\alpha-\beta$ in terms of $a$, $b$, and $c$? Well, we may write $$(\alpha-\beta)^2=(\alpha+\beta)^2 -4\alpha \beta$$ ...
0
votes
1answer
14 views

solve $k(k-1) \geq \ln2*2m$ for k

My Question is related to the birthday problem. Starting at $e^{-\frac{k(k-1)}{2m}} \leq 0.5$ i used $ln(x)$ on both sides and multiplied by $-2m$ to get $k(k-1) \geq \ln2*2m$ According to my ...
7
votes
2answers
151 views

How to find the inverse arc in the configuration space

The following Figure shows the function from configuration space (Torus) to operational space (Annulus). There is a naturally defined continuous function from configuration space $(\theta_A, ...
1
vote
0answers
28 views

How can I solve this para Paradox? [duplicate]

How can I solve this para Paradox? $ -1={(-1)}^{1/2} {(-1)}^{1/2}={[(-1)(-1)]}^{1/2}=1$
0
votes
1answer
362 views

What will be the multiplicative inverse of square root of 5 with respect to a natural number $M$?

Can such a number $N$ be found such that $\sqrt{5}N \equiv 1 \mod M$? If no,what can be the best approximation for $N$?