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
votes
2answers
10 views

Offset a range of numbers

I want to find an equation to offset a range of numbers by a given amount. I'm not sure if I am using the term offset correctly. Lets say the range is from 0 - 1 and I want it to be offset by .25 : 0 ...
2
votes
1answer
16 views

Confusion with modeling a trigonometric function

I am studying trigonometry on Khan Academy and came across this problem: The daily low temperature in Guangzhou, China, varies over time in a periodic way that can be modeled by a trigonometric ...
2
votes
2answers
48 views

How to get $\sqrt {k} + \frac{1}{\sqrt{k+1}}$ in the form $\frac{\sqrt{k^2} + 1}{\sqrt{k+1}}$?

I was wondering if it is possible to get $\sqrt {k} + \dfrac{1}{\sqrt{k+1}}$ in the form $\dfrac{\sqrt{k^2} + 1}{\sqrt{k+1}}$, and if so, how? I ask this, because I'm following this answer, and I get ...
1
vote
1answer
29 views

Is it possible to parametrise $x^{\frac{1}{x-1}}=y^{\frac{1}{y-1}}$?

I don't know if there is a process for parametrising $y^\frac{1}{y}=x^\frac{1}{x}$ and suspect it is not possible to do so. But if it is possible, is it also possible for the similar ...
3
votes
2answers
25 views

What justifies $(\frac{1}{2i})^{n-1}=(-\frac{i}{2})^{n-1}$?

A solution I'm looking at includes the line: $(\frac{1}{2i})^{n-1}=(-\frac{i}{2})^{n-1}$ I'm just not seeing what algebra justifies this...
1
vote
2answers
39 views

Pre-Calc - find the height of triangle

I need help on a problem. I'm lost and I don't know what to do. Could anyone give me some pointers in the right direction? My problem is below: edit - Even though it says test on the top, it isn't. ...
4
votes
1answer
35 views

Find f(x) for an arbitrary x

I try using substitution where $u = \frac{x-1}{x}$ but I just get confused later when doing the algebra. Here's the question. What's the simplest way to solve it? Let $f$ be a function that ...
0
votes
4answers
65 views

Is $g(x) = |x^3| + 1$ even, odd, or neither? [closed]

What is the answer to this? Is $g(x) = |x^3| + 1$ even odd or neither?
0
votes
1answer
22 views

How to add/subtract complex rational expressions?

I'm studying for my Precalculus final and have noticed I still don't fully grasp performing basic operations on complex rational expressions, or finding if any values must be restricted from the ...
0
votes
2answers
28 views

Quadratic Equations GRE Quants

It would be very useful if someone can give me an answer to this question with a proper explanation. One of the factors of the equation $x^2 +9x + c$ is $(x+11)$, where $c$ is a constant. Which of ...
17
votes
3answers
2k views

Monstrous Diophantine Equation

If $x,y\in\mathbb{Z}^+$, then find all the integral solutions to: $$x^6-y^6+3x^4y-3y^4x+y^3+3x^2+3x+1=0$$ I tried solving this question for an hour but still couldn't get it. I tried mod ...
0
votes
1answer
25 views

Convergence/ divergence tests

If i split this equation to test for convergence/divergence, I get one part to be divergent and the other convergent. Can I say something meaningful about convergent + divergent = ? Or is there ...
1
vote
0answers
44 views

Find all the triangles satisfying $\cos(A)\cos(B)+\sin(A)\sin(B)\sin(C)=1$ [duplicate]

I am trying to solve the problem of finding all triangles with angles $A$, $B$ and $C$ (in $[0,\pi]$) such that $\cos A\cos B+\sin A\sin B\sin C=1$. In the case where the triangle has a right angle, ...
-2
votes
1answer
32 views

How many adults went to the park? [closed]

In a school trip to the park, $40$ people went, the adults paid $ 20$ dollar each and the kids paid $10$ dollar each, how many adults went if in total they paid $550$?
1
vote
2answers
25 views

Absolute value of addition of positive real numbers great than that of subtraction?

$$∀a,b ∈ R+, |a + b| > |a - b|$$ I'm wondering if this is true? I'm not sure exactly how I could check or prove it to myself with the absolute value there. I thought I might be able to do ...
0
votes
1answer
23 views

System of linear equations problem - have no idea how to set up equations

The area of a rectangle is 9 more than its perimeter. The length is 3 greater than 4 times the width. What is the area of the rectangle? The answer says the area is 45 square units, and the length is ...
3
votes
2answers
34 views

Graph of $f(x)$ given, find graph of $f(|x|)$

I know the graph of $f(x)=x^2-2x$. Google calculator https://www.google.com/#q=graph+of+x%5E2-2x But how can I find the graph of $f(|x|)=|x|^2-2|x|$? What is the best method to approach here? ...
0
votes
1answer
22 views

How do you solve an equation that has one variable term on one side and constant terms on the other side?

For example, how can it be possible to solve $4+6=2y$? I think I know two ways to do it: $$4+6=2y$$$$4+6-4=2y-4$$$$6=2y-4$$$$6+4=2y-4+4$$$$10=2y$$$${10\over 2}={2y\over ...
1
vote
4answers
104 views

Is $\frac{5x}{3}$ The Same As $\frac{5}{3}x$?

I believe they are the same but I'm not sure. Can someone please clarify this for me, and also explain why it would be the same or different.
1
vote
2answers
44 views

Difficult but Interesting Inequalities Problems

1.) Consider the identity $$(px + (1-p)y)^2 = Ax^2 + Bxy + Cy^2.$$ Find the minimum of $\max(A,B,C)$ over $0 \leq p \leq 1$. 2.) Let $n$ be a positive integer. Show that the smallest integer ...
0
votes
1answer
22 views

Solving a binomial when one of the terms is in the form $e^x$

Say I have the function $y=4e^{-2x}-3x$. I can use a graphing calculator to approximately determine the roots, but how do I find an exact answer?
1
vote
2answers
48 views

The tangent to $1/x$ forms a triangle

Problem: The tangent to $f(x)=1/x$ forms a triangle with the x-axis and the y-axis. Find the area of this triangle. Attempt: I pick some x value $x_0$ which gives me a y value $1/x_0$. So I have one ...
2
votes
2answers
61 views

Let $z_1,z_2$ be complex numbers such that $Im(z_1z_2)=1$ Find the minimum value of $|z_1|^2+|z_2|^2+Re(z_1z_2)$

Question : Let $z_1,z_2$ be complex numbers such that $Im(z_1z_2)=1$ Find the minimum value of $|z_1|^2+|z_2|^2+Re(z_1z_2)$ I know that $|z_1+z_1| \leq |z_1|+|z_2|$ Also if I consider two ...
0
votes
2answers
26 views

Deriving difference from difference of logarithms

Good afternoon. I know that $\log{x} - \log{y} = -0.204$. How do I compute $x - y$? Thanks a lot for your solutions!
0
votes
2answers
55 views

Depressed cubic equation, del Farro's calculation

I am reading about the solution for depressed cubic at http://fermatslasttheorem.blogspot.ca/2006/11/depressed-cubic.html One thing I didn't quite understand is at step 3: if $(3uv + b) = 0$ ...
6
votes
5answers
93 views

Topic for a lecture intended for High School students [duplicate]

I am not sure if this is the right place to post this, but here is the situation. In about two weeks or so I will be giving a 2-3 hours lecture on some topic in mathematics to freshman and sophomore ...
4
votes
3answers
201 views

Why do I get two different results for the reciprocal of $i$?

I am aware that the correct answer is $$\frac{1}{i}=\frac{1}{i}\frac{i}{i}=\frac{i}{i^2}=\frac{i}{-1}=-i$$ But equally, I find no error here: $$\frac{1}{i}=\frac{1}{\sqrt{-1}}= ...
-2
votes
1answer
59 views

who can help me with modeules? [duplicate]

does not work, I've tried a lot of times. need to simplify the phenomena with modules
0
votes
1answer
16 views

Existence of a specific invertible matrix

I am an homework question from course in linear algebra, which I don't know how to solve. I need to know if $\exists A_3$, A is a invertible matrix that holds $A^2=-I_3$ , when I is the identity ...
0
votes
2answers
62 views

No. of real roots of $2^x = 1+x^3$

No. of real roots of $2^x = 1+x^3$ $\bf{My\; Try::}$ Let $f(x)=2^x-x^3-1\;,$ Then $f'(x)=2^x\cdot \ln(2)-3x^2$ and $f''(x)=2^{x}\cdot (\ln 2)^2-6x$ and $f'''(x)=2^x\cdot (\ln2)^3-6$ and ...
0
votes
3answers
39 views

Simplify $\frac {3^{(-3+x)}6^{(3-x)}}{3\cdot4^x}$ [closed]

$$\frac {3^{(-3+x)}6^{(3-x)}}{3\cdot4^x}$$ What is the simplest form?
0
votes
1answer
35 views

Value of indeterminate form — $a_n \to \infty \wedge b_n \to 0$, $\lim_{n\to\infty}a_n\cdot b_n = ?$

$A_n$ and $B_n$ are sequences and $B_n\to 0$ and $A_n\to\infty$. $\lim_{n\to\infty}A_nB_n$ should be equal to $0$ OR $+\infty$ OR $-\infty$? I need to answer yes/no about this problem. I know the ...
0
votes
0answers
13 views

Cuboid with natural number diagonals

I was trying to solve the unsolved problem of finding a cuboid with natural no. sides, face diagonals and space diagonal just as a pastime. I came across the following question. $A=(m-n)(x-y)$ ...
0
votes
5answers
64 views

show that $(1+x^2)(1+x^4)(1+x^8)\cdots (1+x^{2^n}) = \frac{1-x^{2^{(n+1)}}}{1-x^2}$

I am trying to solve the following question in my textbook, one way to go at this would probably be to use induction to prove the statement. But I am looking for alternativ ways to prove this. ...
6
votes
4answers
555 views

Finding polynomials with their values at points

Is there any way I can find a polynomial given any 2 points (with x coordinate OF MY CHOICE): Let's say there's some polynomial I don't know(p(x)=2x3+x2+3), but my machine will give me an output. I ...
2
votes
1answer
43 views

$f(x)$ be a polynomial with integer coefficients and $f(0) = 1989$ and $f(1) = 9891$, then no. of polynomial

Let $f(x)$ be a polynomial with integer coefficients and $f(0) = 1989$ and $f(1) = 9891$. Then prove that $f(x)$ has no integer roots. $\bf{My\; Try::}$ Let $f(x) = ...
0
votes
1answer
21 views

Solving a system of complex equations

$$u = (1+i, i), v = (1-i, 2i), w = (2,3+i)$$ I'm asked to find is there's $z$ such that: $$v = zu$$ So if I suppose $z = a+bi$ I have the system: $$(1-i, 2i) = (a+bi)(1+i, i)\implies\\(1-i, 2i) = ...
0
votes
3answers
32 views

Why does $\frac{1}{4}x^2 + \frac{1}{2} + \frac{1}{4x^2} = (\frac{1}{2}x + \frac{1}{2x})^2$

$\frac{1}{4}x^2 + \frac{1}{2} + \frac{1}{4x^2} = (\frac{1}{2}x + \frac{1}{2x})^2$ This is part of a solution to a more complex problem. Can someone explain what method was used here and how it works? ...
0
votes
4answers
43 views

Finding the $x$-intercept when variable has fractional exponent

The equation is $$2x-3x^{\frac 23}+4 = 0$$ How would one go about finding the x-intercept(s) of this equation? I have tried, but am unable to isolate the $x$. EDIT: Changed from g(x) = expression ...
2
votes
2answers
75 views

A positive integer $n$ is such that $1-2x+3x^2-4x^3+5x^4-…-2014x^{2013}+nx^{2014}$ has at least one integer solution. Find $n$.

A positive integer $n$ is such that $$1-2x+3x^2-4x^3+5x^4-...-2014x^{2013}+nx^{2014}$$ has at least one integer solution. Find $n$.
1
vote
1answer
62 views

Find consumer demand as a function of time, given the demand equation and price

An importer of Brazilian coffee estimates that local consumers will buy approximately $Q(p)= 4374/p^2$ kg of the coffee per week when the price is $p$ dollars per kg. It is estimated that $t$ weeks ...
2
votes
3answers
34 views

A polynomial $f(x)$ and its behavior as $f(t)>5$

Let $f(x)$ be a polynomial with integer coefficients. Suppose there are four distinct integers $p,q,r,s$ such that $f(p) = f(q) = f(r) = f(s) = 5$. If $t$ is an integer and $f(t)>5$, what is the ...
3
votes
3answers
61 views

Proving $x^2 - y^2 + z^2 \gt (x - y + z)^2$ [closed]

Prove that $$x^2 - y^2 + z^2 > (x - y + z)^2$$ where: $x < y <z$ for all natural numbers. Thank for help.
6
votes
3answers
48 views

If $f(x) $ and g(x) are functions such that $f(x+y) =f(x)g(y) +g(x) f(y) $ then …

Question : If $f(x) $ and g(x) are functions such that $f(x+y) =f(x)g(y) +g(x) f(y) $ then $\begin{vmatrix} f(\alpha) & g(\alpha) & f(\alpha + \theta) \\ f(\beta) & g(\beta) & f(\beta ...
0
votes
0answers
52 views

Multi-ruled combinatorics problem (need this for my lab)

I need to know this for practical purposes and not homework, learning etc.. Say I have 3 electrodes A,B and C. Say I also have 3 electrolytes A,B and C. If electrode A has to be in electrolyte A, ...
3
votes
2answers
30 views

Function notation meaning: $f: \{a,b\} \to a$ - Zorich - MA I - p18

I have some notation I haven't seen before: $$f: \{a,b\} \to a\text{ and } g:\{a,b\}\to b$$ What does this mean? We are mapping from some $X=\{a,b\}$ to some $Y=a$? So pretty much we are always ...
1
vote
0answers
45 views

Which math class next

I just finished and Algebra for Calculus class this semester. I'm trying to work up to taking calculus (have to do up through calc 3). One person told me I should take trig next, and another calculus. ...
0
votes
0answers
31 views

What is the point of reflection of this function

$$y = 3x(x+5)^{2/3}$$ Is there some kind of trick to simplify it?
0
votes
2answers
19 views

How can a given length of something yield different sum in square meters?

How can a rope of say 100 meters yield different return in square meters, based on how you divide each side? E.g. 10m x 10m = 100m2 15m x 5m = 75m2 Now of course I see that based on how you choose ...
0
votes
2answers
38 views

When looking for zeros of a rational function, why is the numerator equated to zero and not the denominator?

If you have a function $F(x)=\dfrac{a(x)}{b(x)}$ and you are asked to find the zero(s) of the function, why do you set the numerator equal to zero, and not the denominator?