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.

learn more… | top users | synonyms (2)

0
votes
3answers
42 views

If $f(ab)f(ac)f(bc)f(a+b)f(a+c)f(b+c)=2015$ for every positive (non-zero) $a$, $b$ and $c$, find $f(2016)$.

If $f(ab)f(ac)f(bc)f(a+b)f(a+c)f(b+c)=2015$ for every positive (non-zero) $a$, $b$ and $c$, find $f(2016)$. Can someone help me?
1
vote
2answers
44 views

Showing that $\alpha \beta$ is the root of a polynomial

Assuming that $\alpha, \beta$ are distinct roots of $P(x) = x^4+bx^3-1 = 0$, where $b \in \mathbb R$, show that $\alpha \beta$ is the root of $Q(x) = x^6+x^4+b^2x^3 -x^2 -1$. I have already noticed ...
0
votes
1answer
19 views

Find the sum of the squares of all sides and diagonals of a n-gon inscribed in a circle.

With a circle with radius r and center A, for any homogenous n-gon -- find the sum of the squares of all sides and diagonals of the n-gon inscribed within the circle. I believe the general rule for ...
0
votes
1answer
39 views

If $a^{1/x}=k$ then how is $a=k^x$? [on hold]

If $a^{1/x}=k$ then how is $a=k^x$? It's a basic thing but I'm having a little problem understanding this thing.
1
vote
0answers
25 views

Solve these simple simultaneous equations?

Assuming $x_1, x_2 \geq 0, \lambda \neq 0, w_1,w_2 > 0$ We have the equalities: $$w_1 - \lambda x_2 = 0 ... (1)$$ $$w_2 - \lambda x_1 = 0 ... (2)$$ $$\bar y - x_1x_2= 0 ... (3)$$ My solutions ...
2
votes
1answer
8 views

Solve for a hyperbolic Laplace Transform by expressing as exponents and shiftig on s-axis (5.3-21)

I cannot get past a certain point on this problem as shall be shown. I need guidance in order to complete the problem. The exercise as stated in the text: Represent the hyperbolic function in terms ...
0
votes
2answers
48 views

If $2x = a + b + c$, show that $(x − a)^2 + (x − b)^2 + (x − c)^2 + x^2 = a^2 + b^2 + c^2$ .

Having trouble solving this. If $2x = a + b + c$, show that $(x − a)^2 + (x − b)^2 + (x − c)^2 + x^2 = a^2 + b^2 + c^2$. .
0
votes
2answers
34 views

Re-arranging an equation help

How do I re-arrange the equation $$ -150 = (-9.8)(t) + 0.5(-9.8)(t)^2 $$ and solve for $t$? I collected the like terms firstly, so $$-150 = -48.02 \cdot t^3$$ then I knew I was doing something ...
2
votes
2answers
31 views

given a circle $(x-1)^{2}+ y^{2}=1$, find $b$ such that the line $y=x+b$ intersects with the circle just once.

given a circle $(x-1)^{2}+ y^{2}=1$, find $b$ such that the line $y=x+b$ intersects with the circle just once. This question is for a precalculus class so setting the derivative of the positive ...
2
votes
2answers
156 views

Solving a second-degree exponential equation with logarithms

The following equation is given: $8^{2x} + 8^{x} - 20 = 0$ The objective is to solve for $x$ in terms of the natural logarithm $ln$. I approach as follows: $\log_8{(8^{2x})} = \log_8{(-8^{x} + ...
1
vote
3answers
29 views

Problem with simultaneous equations

Given that $(5, h)$ is a solution of the simultaneous equations $h(x-y) = x + y -1 = hx^2 - 11y^2$, find (a) the value of $h$. (b) the other solution of the equation.(x and y) I don't ...
1
vote
1answer
44 views

Precalculus - connect 2 towns

A state highway department plans to construct a new road between towns $A$ and $B$. Town $A$ lies on an abandoned road that runs east-west. Town $B$ is $20$ miles north of the point on that road that ...
1
vote
1answer
31 views

How to simplify logs and powers?

Is there any way to simplify $(\log a)^{\log b} = c$? And even this $(\log x)^y = z$? And also this $(\log m)(\log n) = p$ (which is essentially $\log m^{\log n} = p$) I was trying to simplify some ...
0
votes
2answers
24 views

Find the Domain and Sketch the Graph of the Function $h(x)= \frac{3x+|x|}{x}$

\begin{align*} h(x) & =\dfrac{3x+|x|}{x}\\ & = \begin{cases} \dfrac{3x+|x|}{x} & \text{if $x > 0$}\\ \dfrac{3x + |-x|}{x} & \text{if $x < 0$} \end{cases} \end{align*} I ...
0
votes
1answer
56 views

$a,b,c$ are integers ,and $ \frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ ,$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ are integers, prove $a=b=c$

Here is my solution; can someone check it out? The sum $$ \frac{b}{a}+\frac{c}{b}+\frac{a}{c} $$ is just $$ \frac{a}{b}+\frac{b}{c}+\frac{c}{a} $$ but 'reversed'. No element of this expression can ...
1
vote
1answer
40 views

$ab-(a+b)(a-b)=0$ and the Golden ratio.

I have found: $b=a*\phi$ $b=a*(-\phi)$ $b=a/\phi$ Trying to find the correlation with the equation and phi, any insight how to demonstrate this or a proof?
1
vote
2answers
32 views

Given some of the roots of the function $f(x) = x^3+bx^2+cx+d$, how do I find the coefficients of that function?

Two of the roots of $f(x) = x^3+bx^2+cx+d$ are $3$ and $2+i$. How do I find b+c+d? The answer choices are -7, -5, 6, 9, and 25.
0
votes
1answer
33 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 ...
1
vote
1answer
23 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 ...
3
votes
1answer
83 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
2answers
33 views

variables to the power of a fraction

I have this question for advanced math, I can't seem to get my head around. $$\frac{x^{5/2}}{(x^{1/3})^4}$$
0
votes
7answers
665 views

Is it possible to find the product of two numbers given their difference?

That is, find $a\cdot b$ given the value of $a-b$. Is it possible?
0
votes
1answer
40 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
11 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
19 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!
0
votes
0answers
19 views

solve this equation $1 = \left( \frac{\mu}{f} \right)^{\frac{3}{2}} \left( 1+ \frac{ \pi^2}{8} \left( \frac{kT}{\mu} \right)^2 \right)$

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 ...
20
votes
7answers
1k 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, ...
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
2answers
25 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 = ...
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 ...
3
votes
4answers
481 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}+ ...
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 ...
-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
29 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 ?
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, ...
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
50 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
36 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
51 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 $n = 2$.
4
votes
0answers
58 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 ...
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 ...
0
votes
1answer
53 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 ...
-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, ...
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$
-7
votes
4answers
98 views

Can $9+10$ equal $21$? [duplicate]

I just saw a video with the following: $9+10=21$ because $$9 = 3*3$$ $$10 = 5*2$$ if $$5*3=15$$ and $$2*3=6$$ $$15+6=21$$ does that not prove that $10+9=21$?
0
votes
1answer
31 views

Which functions are power functions? [on hold]

I have 6 functions and want to know which ones are power functions and which ones aren't. I know that the power function has the form $f(x)=Kx^p$, where $K$ and $p$ are constants. $f(x)= \pi*x^4$ ...
-1
votes
2answers
20 views

Find closed form for a sequence using the Fibonacci and Lucas number sequences. [on hold]

Define $A_n$ as follows: $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that ...