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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1answer
36 views

Does this loop run in $\mathcal{O}(n^4)$ time?

A double loop is given: int sum = 0; for (int i = 0; i < N*N; i++) for (int j = i; j < N; j++) sum++; My analysis: The inner loop runs $n$ ...
1
vote
2answers
28 views

compostition of functions

$f(x)=12(8x+3)$ $h(x)=(3x+19)−5$ Solve when $x=8$ I really need step by step directions to solve $f(h(x))$. Also can someone solve $h(f(x))$. I really need help this is so confusing. I am able to ...
28
votes
5answers
5k views

Trick with 3-digit numbers, always get 1089

When I was in primary school a teacher showed us the following exercise in arithmetic. Take any 3 digit number between 201 and 998 provided that the hundreds digit is bigger than the ones digit and ...
0
votes
1answer
45 views

An identity that is always an integer

If $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \in \mathbb{Z}$$ and $$\frac{b}{a} + \frac{c}{b} + \frac{a}{c} \in \mathbb{Z}.$$ For any natural $n$, is the following true? $$\frac{a^n}{b^n}+\frac{b^n}{...
2
votes
1answer
76 views

Why do we say that $\sqrt{-0} = -0$?

According to wikipedia's page on signed zeroes, we agree that $\sqrt{-0} = -0$. I would always have guessed that it would be $0i$, as $(0i)^2 = 0^2*i^2 = 0 * (-1) = -0$. I know that my own ...
0
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2answers
75 views

Is my method of computing the running time correct?

Okay, so this is the code for which I need to compute the running time: ...
7
votes
4answers
223 views

How do you read the symbol “$\in$”?

A variable in an equation may be replaced by any of the numbers in its domain. The resulting equation may be either true or false. Here is another way to show ...
0
votes
1answer
44 views

What does the Fundamental Theorem of Algebra say about…

the number of complex zeros of a polynomial function? would it be safe to say it states the complex zeros of a polynomial function always come in pairs where they are conjugates of each other... or ...
1
vote
1answer
154 views

What is the value of $\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$? [duplicate]

How to compute $$S=\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$$ I tried to rewrite it in terms of $\sin$ $$ \csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}...
1
vote
4answers
416 views

What does the Fundamental Theorem of Algebra say about the number of complex zeros of a polynomial function?

I was watching the Khan Academy video on the Fundamental Theorem of Algebra when I got confused by something that Sal Khan states. From what I understand, the Theorem says that the complex zeros of a ...
0
votes
1answer
21 views

Graphing functions error

please check my answers. Why am I getting these functions of graphs wrong? http://i.imgur.com/SfdeGOn.png?1 and http://i.imgur.com/o7PmcQY.png?1 Thank You
0
votes
1answer
17 views

Rewriting the difference of two $3/2$-powers

There's this part in this problem where it goes $\frac{8}{27}\left[\left(\frac{22}{4}\right)^{3/2} - \left(\frac{13}{4}\right)^{3/2}\right]$ and it equals $\frac{22\sqrt{22} - 13\sqrt{13}}{27}$. If ...
0
votes
4answers
1k views

Find all complex and real roots of higher degree polynomials, given one root

$2+3i$ is a zero of $f(x)=x^4-4x^3+17x^2-16x+52$, find all of the zeros of $f(x)$ thanks!
1
vote
2answers
59 views

finding solutions by factoring

How would you find the integer solutions to $a^2-b^2=16$? I know that the factors of $16$ are $8*2,$ $4*4,$ and $16*1.$ How would I use this? I know that $a^2-b^2=(a+b)(a-b)=16,$ but how would you ...
0
votes
2answers
33 views

replacing numbers to get final anser

I found this question in a random problem solving book that I was reading and wanted to know how you would solve it. I am not sure as how to go about this. Take any positive integer $n$ with fewer ...
3
votes
1answer
357 views

How to get the ratio from a function of N?

The exercise gave us a chart which showed the running time as a $N$ increases: \begin{array}{c|c} N & \text{seconds}\\\hline 256 & 0.000\\ 512 & 0.000\\ 1024 &...
0
votes
1answer
32 views

Help symplifyinf this pre-calculus queston please

$$\frac{ab}{a^2+ab+b^2}+\left(\frac{ac-ad-bc+bd}{ac-ad+bc-bd}\cdot\frac{a^3+b^3}{a^3-b^3}\right)$$ I am stuck trying to simplify this expression, I need help please.
2
votes
1answer
36 views

How to isolate x in this case?

We have the following equations: $x^3 + px + q = 0$ $x = u + v $ $p=-3uv$ Where $q$ and $p$ are known real numbers and $x$ is an unknown real number. We want to find an equation for $x$...
0
votes
2answers
64 views

finding integer solutions for a and b

Show that the only positive integer solutions for $a$ and $b$ in the equation $a^2-b^2=16$ are $a=4, b=0$ and $a=5, b=3.$ How many pairs of solutions would there be if we allowed negative values for ...
5
votes
1answer
114 views

At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
2
votes
1answer
25 views

Find an equation for u and v

We have the function $x^3 + px + q = 0$, where $p$ and $q$ are known real numbers and $x$ is an unknown real number. Put $x = u + v$ and write it out. If $3uv+p=0$, can you find another equation ...
0
votes
2answers
39 views

finding values of $x$ in $Z$

Find all values of $x$ such that $\frac{x-4}{2x-3}\in\mathbb Z$? I came up with this question to see if it could be solved based on some other questions I did myself. I thought this could not be ...
1
vote
0answers
36 views

Find the reflection point $P$

On the real number line, paint red all points that correspond to points of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer points blue. Find a point $P$ on ...
7
votes
1answer
258 views

How to represent Fermat number $F_n$ as a sum of three squares?

Let $F_n=2^{2^n}+1$ be the Fermat number. How to represent the Fermat number $F_n$ for $n \geq 3$ as a sum of three squares of different natural numbers? For example for $n=3$ we have $$ F_3=...
5
votes
4answers
116 views

Rationalize $\left(\sqrt{3x+5}-\sqrt{5x+11} -\sqrt{x+9}\right)^{-1}$

I was trying to find if there a method similar to multiplying and dividing by the conjugate $$\frac{1}{\sqrt{3x+5}-\sqrt{5x+11} - \sqrt{x+9}},$$ but that doesn't seem to work here. Also, is there a ...
3
votes
2answers
76 views

Why this gamma function reduces to the factorial?

$$\Gamma(m+1) = \frac{1\cdot2^m}{1+m}\frac{2^{1-m}\cdot3^m}{2+m}\frac{3^{1-m}\cdot4^m}{3+m}\frac{4^{1-m}\cdot5^m}{4+m}\cdots$$ My books says that in a letter from Euler to Goldbach, this expression ...
0
votes
1answer
24 views

SAT Math Problem about decimal

In the decimal representation of $\frac{1}{k}$, where $0 < \frac{1}{k} < 1$. the tenths digit is $1$, hundredths digit is $3$ and at least one other digit is nonzero. What is the tenths digit ...
1
vote
1answer
23 views

Help with a graph

I need help with the graph of: $x^2+y^2=|x|+|y|$ I tried squaring both sides and I got $x^4+2x^2y^2+y^4-x^2-2xy+y^2=0$ but still can't figure the graph.
0
votes
2answers
70 views

Prove by Simple Induction that $12^n − 1$ is divisible by $11$ for each $n \in \mathbb N.$

Since $12^n-1$ is divisible by $11$ for small $n$ cases i.e. $(1,2,3,\ldots$, etc), I want to prove that $12^{n+1} -1$ is also divisible by $11$. what I wrote down: ...
0
votes
3answers
37 views

Determining the composition $f(h(x))$

$f(x)= 12(8x+3)$ $h(x)= (3x+19)-5$ I really need step by step directions to solve $f(h(x))$. Also can someone solve $h(f(x))$. I really need help this is so confusing. I am able to do the simpler ...
1
vote
0answers
66 views

Determine whether $f :\mathbb R\rightarrow \mathbb R$ given by f(x)=$x^3+1$ is surjective or not.

1) Determine whether the function $f:\mathbb R\rightarrow \mathbb R$ given by $f(x)=x^3+1$ is surjective or not. Explain your answer carefully with precise arguments. Solution This function is ...
0
votes
1answer
26 views

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

Do I take the x out of the absolute value sign? If so $ \frac{x(3+1)}{x} = 3+1$, thus, $4$ Which would only be a point. Am I correct?
1
vote
1answer
26 views

Solving an equation with exponents by using logarithms

Solve the equation $$0.25^5 = 4^{(5x-3)/3} \cdot (0.125)^{6x}$$ So would I just bring down the exponents by taking the log of each constant?
0
votes
1answer
23 views

Find the domain of $h(x) = 1 /\sqrt[4]{x^2-5x}$

I need to find the domain of $$h(x) = \dfrac 1{\sqrt[4]{x^2-5x}} $$ I took $x^2-5x$ and set it $>0$ $x^2 > 5x$ I am stuck at this point I thought I could factor the x out and got $x(x-5) >...
0
votes
1answer
23 views

Find the Domain of the function: $f(x) = \frac{x^2+2x+3 }{ x^2 -9}$

My answer is all reals for the numerator and $(-∞, -3) \cup (-3, +3) \cup (3,∞)$. Am I correct?
0
votes
1answer
283 views

Log arithmic Equation - Graph curved line

I'm recreating the graph picture below with equations. Using the online graphing tool "Desmos": These are all the equations I have done so far, with there restrictions top stop at specific points. ...
1
vote
3answers
65 views

PEMDAS question: $F(x) = 3x^2 - x+2$. Find $[f(a)]^2$

How should I go about doing this? $(3a^2-a+2)^2$? Thus, $9a^4-a^2+4$
2
votes
1answer
397 views

Solving the recursion $y_n=2ny_{n-1}$ with Wolfram|Alpha

This is blowing my mind away ... this should be easy stuff! Starting with the recursive formula $y_n=2n*y_{n-1}$ where $y_1=5.$ I'm trying to come up with a formula for the series { 5, 20, 120, 960, ...
2
votes
2answers
29 views

Solve the algebra equation- unsure about order of operations, how to go about solving, solve for x

The question states: solve the equation. State the solution set and check your answer. I've spent a good 45 minutes on this, to know avail. If someone could sort of walk me through this I would be ...
0
votes
3answers
56 views

Why is the absolute value of the | √2-1| = √2 - 1 and why is the |3-π| = π-3?

I understand that the absolute value of a number |a|, is the distance from a to o on the real number line, and that distances are always positive or zero, so why is |√2-1| = √2 - 1? Shouldn't it be √...
1
vote
2answers
23 views

Compound absolute value with different left and right values

I'm not sure how to find the solution set for a compound absolute value with different left and right values. Here is an example: A = { 2 ≤ |x| < 4 : x ε integers } My thinking is to create two ...
1
vote
2answers
560 views

Polynomial Function application

The rabbit population on a small island is observed to be given by the function $$P(t)=120t-0.4t^{ 4 }+1000$$ where $t$ is the time (in months) since observations of the island began. When is the ...
1
vote
3answers
24 views

Question about upper bounds

If $a$ is an upper bound for the real zeros of the polynomial $P$, then $-a$ is necessarily a lower bound for the real zeros of $P$. Is this true? I think it is false because if we divide $P(x)$ by $...
0
votes
1answer
44 views

solve an quadratic equation

I was reading a document , where I stucked in figuring out this equation. $f(k)= k^2-nk+\frac{n^2 - n}{2}$. This is a quadratic function of $k$. It is minimized when $k=\frac{n}{2}$ (the $k$ ...
2
votes
1answer
81 views

If $\sin \phi$ and $\tan \phi$ are the roots of the equation $ax^2+bx+c=0$, compute $b^2-c^2$

If $\sin \phi$ and $\tan \phi$ are the roots of the equation $ax^2+bx+c=0$. Then $(b^2-c^2) = $ $\bf{Options::}$ $(a)\;\; 4ac\;\;\;\;\;\;(b)\;\; a^2\;\;\;\;\;\;(c)\;\; 4bc\;\;\;\;\;\;(d)\;\; 4ab$ ...
2
votes
1answer
66 views

Find the minimum value of $x+y+z.$

Let $x,y,z$ are nonegative such that $(x - y)(y - z)(z - x) \geq 1.$ Find the minimum value of $x+y+z.$
1
vote
3answers
62 views

Solving for $x$ given $y = 2x - 9$ and $y = 5$. When does the 5 come into play?

I am doing some problems. It states "Isolate the left side for each of the following equations. Then solve for x, assuming the value of y is 5 in a all cases." I noticed that I can get two different ...
-1
votes
2answers
45 views

Find the value of $3qx^2-2px+3q$ [closed]

If $\frac{\sqrt{p+3q}+\sqrt{p-3q}}{\sqrt{p+3q}-\sqrt{p-3q}}=x$ then,find the value of $3qx^2-2px+3q$.
1
vote
3answers
136 views

Calculate the depth of water in the trough when it is exactly half full

I am in my last year of high school and am currently studying for my finals by going over exercises in my coursebook. I came across this exercise and have been stuck on it for some weeks now. I have ...
2
votes
2answers
119 views

What is $\frac{2x}{1-x^2}$ when $x=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$?

If $$x=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$$ Find $$\frac{2x}{1-x^2}$$ I got till here by simplification by taking the previous value of x, ie, $$x={\frac{\sqrt{1-\cos\theta}}{1+\cos\theta}}$$...