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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3answers
40 views

Multiplicity of a root of a polynomial

:) It's true that, if a polynomial has a root (let's say, k, for example) with multiplicity n (n>1, for n integer), then it's true that the derivate polynomial have k as a root with multiplicity ...
5
votes
2answers
35 views

Doubt with Absolute Value Inequality

Problem: Find all values of $x$ for which $\dfrac{|x-2|}{x-2}>0$ My incorrect attempt: Using the definition the Modulus, $|x-2|=x-2$ for all $x\ge2$ and $|x-2|=-x+2$ for all $x\le2.$ ...
4
votes
1answer
70 views

If $x=(9+4\sqrt{5})^{48}=[x]+f$ . Find $x(1-f)$.

If $x=(9+4\sqrt{5})^{48}=[x]+f$, where $[x]$ is defined as integral part of $x$ and $f$ is a fraction, then $x(1-f)$ equals . $\color{green}{a.)\ 1} \\ b.)\ \text{less than}\ 1 \\ c.)\ ...
-4
votes
3answers
72 views

The limit of iterated square root with multiplication under the root, $\sqrt{ a \sqrt{ a \sqrt{a \cdots}}}$ [duplicate]

$$ \sqrt{ a \sqrt{ a \sqrt{a \cdots}}}=\text{ ?} $$ options were given as $0$ $-a$ $a$ $1$ i did not know how to solve it or what it was related to. Could anyone please explain the concept ...
-3
votes
1answer
31 views

Coding Matrix Problem

(A) Use the coding matrix $A=\left[\begin{smallmatrix}1&3\\5&-3\end{smallmatrix}\right]$ to encode the word jump (B) Using it's Inverse $A^{-1}= ...
0
votes
2answers
28 views

Why does $\sqrt{1 + 4x^2} > 2|x| \implies -1 - \sqrt{1 + 4x^2} < -2x$?

Given $f(x) = \frac{-1 - \sqrt{1 + 4x^2}}{2x}$, if $x > 0$, show that $f(x) < -1$. Solution: Note that $\sqrt{1 + 4x^2} > 2|x|$. So if $x > 0$, then $-1 - \sqrt{1 + 4x^2} < ...
-3
votes
2answers
30 views

want to know about probability? [on hold]

A certain fruit stand sold apples for \$0.70 each and bananas for \$0.50 each. if a customer purchased both apples and bananas from the stand for a total of \$6.30, what total number of apples and ...
-1
votes
2answers
37 views

What is a proof for the product rule for logarithms? [on hold]

What are some proofs for this? I'd like to see more than one type if possible.
-1
votes
6answers
122 views

What is the mathematical symbol for the sum of numbers

For example, when $n=5$, what is the symbol for $5+4+3+2+1$?
-1
votes
1answer
45 views

summation problem [on hold]

what is the result for the following double summation: $\sum\limits_{i \neq j}^{\infty}\alpha^i\alpha^j$ where $ i, j =0,1,2,.......$
0
votes
2answers
41 views

From the graph find the number of solutions.

The figure below shows the function $f(x)$ . How many solutions does the equation $f(f(x))=15$ have ? $a.)\ 5 \\ b.)\ 6 \\ c.)\ 7 \\ d.)\ 8 \\ \color{green}{e.) \ \text{cannot be determined from ...
0
votes
2answers
75 views

Make $\frac{a}{b+c} = \frac{a}{b} + \frac{a}{c}$ right

I have this math exercise where I have to make the equation $$\frac{a}{b + c} = \frac{a}{b} + \frac{a}{c}$$ right. Can anyone help me please? I have already tried things like $$a\left(\frac{1}{b} + ...
1
vote
1answer
80 views

Need hint to solve a nasty integral.

Let $f(x)=\frac{x+2}{2x+3}$, $x>0$. If $$\int \left( \frac{f(x)}{x^2} \right)^{1/2}dx=\frac{1}{\sqrt{2}}g \left(\frac{1+\sqrt{2f(x)}}{ 1-\sqrt{2f(x)}} \right) -\sqrt{\frac{2}{3}}h ...
5
votes
2answers
95 views

Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer

Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer. Doesn't this problem seem a little out of the box? It seems beautiful, but I don't have an idea on how to start. Calculating the value does ...
1
vote
1answer
57 views

Prove $e^x$ limit definition from limit definition of $e$.

Is there an elementary way of proving $$e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n,$$ given $$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n,$$ without using L"Hopital's rule, Binomial Theorem, ...
1
vote
2answers
43 views

Concluding three statements regarding $3$ real numbers.

$\{a,b,c\}\in \mathbb{R},\ a<b<c,\ a+b+c=6 ,\ ab+bc+ac=9$ Conclusion $I.)\ 1<b<3$ Conclusion $II.)\ 2<a<3$ Conclusion $III.)\ 0<c<1$ Options By ...
6
votes
2answers
78 views

Proving that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$.

I need to prove, that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$. Prove that $$x^m+x^{-m}=P_m (x+x^{-1} )=a_m (x+x^{-1} )^m+a_{m-1} (x+x^{-1} )^{m-1}+...+a_1 (x+x^{-1} )+a_0$$ on ...
-1
votes
5answers
101 views

What is the value of $x*y$? [on hold]

Given that $$\left(\frac{x}{y}\right)^{-2} + \left(\frac{y}{x}\right)^{-2} = \frac{10}{3}$$ find the value of $x*y$. My question is, can we calculate the value of $x*y$ or not? If yes, then ...
4
votes
0answers
60 views

Long polynomial expansion with 34 roots

This is a very tricky problem, I just need a few hints. I think the $(-x^{17})$ is also there for a specific trick. In the end if it is $ax^{17}$, I see that $a = 17 - 1 + 1 = 17$. Also, another ...
3
votes
4answers
111 views

Why memorize trig identities?

I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my ...
0
votes
3answers
45 views

How to find the range of the function $\frac{x+2}{x+1}$ with domain $x \geq 0$?

How to find the range of the function $\frac{x+2}{x+1}$ with domain $x \geq 0$? I am a high school student and stuck at this simple question on domains and ranges of functions. I have done the ...
0
votes
1answer
41 views

Finding value of an exponential equation [duplicate]

If $ x^2=3x-1$ then find the value of $(x^6+1)/x^3$ I used the quadratic but it became too complicated
1
vote
5answers
150 views

Finding value of an expression

If $x^2-3x-1=0$ then find the value of $(x^6+1)/x^3$ I tried to solve the quadratic but it became too complicated any way of doing this without a calculator
1
vote
4answers
259 views

Negative exponents and positive numbers.

Why is it that when we raise a positive number to a negative power, we don't get a negative number?
3
votes
2answers
76 views

If $\frac{(b−c)}{a} + \frac{(a+c)}{b} + \frac{(a−b)}{c}=1$ and $a-b+c \neq 0 $, then prove that $\frac 1a = \frac 1b + \frac 1c$

The question given is If $\dfrac{(b−c)}{a} + \dfrac{(a+c)}{b} + \dfrac{(a−b)}{c}=1$ and $a-b+c \neq 0 $ then prove that $\dfrac 1a = \dfrac 1b + \dfrac 1c$ I tried to take $abc$ on the right ...
1
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0answers
51 views

A geometric interpretation of a geometric series

The following code is compiled by TikZ to draw a right triangle so that the enclosed area is partitioned by infinitely many triangles similar to itself. I saw on ...
-1
votes
3answers
33 views

Question in fraction [on hold]

I find it difficult to solve this. Please help me out.... At a party, the number of men was $\frac{2}{5}$ the number of women. After $144$ women left, the number of women become $1\frac{1}{2}$ ...
0
votes
5answers
47 views

Help Solving a Simultaneous Equation.

Im currently doing my Kumon (A math tutoring center I guess) homework, and Im having a bit of difficulty answering a simultaneous equation, involving $x$ and $y$ variables to the second power. School ...
0
votes
1answer
43 views

The value of $x$ for which function attains max value

At what value of $x,\ x\in \mathbb{Z}$ will the function $\dfrac{x^2+3x+1}{x^2-3x+1}$ attain its maximum value . $\color{green}{a.)\ 3 }\\ b.)\ 4 \\ c.) -3 \\ d.)\ \text{none of these} \\ $ ...
5
votes
1answer
71 views

Inequality with reciprocals of $n$-variable sums

Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Is it always true that $$\sum_{i=1}^n\frac{1}{a_i}-\sum_{1\leq i<j\leq n}\frac{1}{a_i+a_j}+\sum_{1\leq i<j<k\leq ...
2
votes
3answers
41 views

Reasoning about numbers close to two other numbers $a,b$ (inequalities)

Let $a < b$ and $0 < \varepsilon < (b - a)$ and let $x, y \in \mathbb R$ be such that $$ | x - a | < \frac{(b - a) - \varepsilon}{2}, \qquad | y - b | < \frac{(b - a) - ...
1
vote
4answers
116 views

Why is $\sqrt{x^2}= |x|$ rather than $\pm x$? [duplicate]

Shouldn't the square root of a number have both a negative and positive root? According to Barron's, $\displaystyle \sqrt{x^2} = |x|$. I don't understand how.
1
vote
2answers
48 views

Find $f\circ f$ if $f(t)=\dfrac{t}{(1+t^2)^{1/2}},\ \ t\in \mathbb{R}$

Find $f\circ f$ if $f(t)=\dfrac{t}{(1+t^2)^{1/2}},\ \ t\in \mathbb{R}$ $ a.)\ \dfrac{1}{(1+2t^2)^{1/2}} \\ \color{green}{ b.)\ \dfrac{t}{(1+2t^2)^{1/2}}}\\~\\ c.)\ (1+2t^2)\\~\\ d.)\ ...
1
vote
5answers
42 views

Proof of the power rule for logarithms

What is the proof for the power rule for logarithms? And are there different ways to prove it?
8
votes
3answers
567 views

e and its applications

In math when people want to model population growth or radioactive decay we use exponential functions. In many cases, we use base $e$. My question is, what is the purpose of using base $e$ rather than ...
13
votes
0answers
95 views
+50

Inclusion-exclusion-like fractional sum is positive?

Let $A_1,A_2,\ldots,A_n$ be finite nonempty sets. Is it true that $$\sum_{i=1}^n\frac{1}{|A_i|}-\sum_{1\leq i<j\leq n}\frac{1}{|A_i\cup A_j|}+\sum_{1\leq i<j<k\leq n}\frac{1}{|A_i\cup ...
22
votes
4answers
789 views

Plot of a … Square?

Well there are equations which can plot a square like : $|x-y|+|x+y|=a$ But how about this equation: ? (At the end ... bear with me!) [Here I have taken $a = 1$] Plot of $$x^2 + y^2 = a^2$$ ...
1
vote
1answer
41 views

(Visual) Intuition: Division and complex fractions

When treating division as "groups of the numerator" (sorry, I don't know the technical term -- see image), why does a complex fraction in the denominator get added together to produce a 1 (number of ...
2
votes
1answer
34 views

How to solve equations containing multiple $|x|$s?

Suppose I have an equation which looks like: $$|x-2| + |2x+1| = 3$$ or, $$|x-1| + |x-3| - |5x-1| = 2$$ How should I solve such problems? What i do is generally a kind of "hit-and-trial" ...
0
votes
2answers
47 views

Graph $y=|x+8|+|x-8|$

Graph $y=|x+8|+|x-8|$ I tried to simply this with $$y=(x+8)+(x-8) \implies y=2x,x>0\\ y=(-x+8)+(-x-8) \implies y=-2x,x<0$$ But this looks quite different from the original. I look ...
0
votes
0answers
29 views

Investment question.

Jack invests \$1000 at a certain annual interest rate, and he invests another \$3000 at an annual rate that is one-half percent higher. If he receives a total of \$135 interest in 1 year, at what ...
0
votes
0answers
39 views

Is there any practical use of this algorithm?

Example The exact solution of a DE $\frac{dp}{dv}=-1.4\frac{p}{v} $ with initial condition $(P_1,V_1)=(1,1)$ can be obtain by solving the integral ...
7
votes
1answer
44 views

Basic absolute value property

Hello all I am wondering if anyone has the correct proof that I should use for Spivak calculus ( chapter 1, question 12 ) that says $$|xy|=|x| \cdot |y|$$ from past times I know it is true , but I ...
-5
votes
1answer
57 views

Subtraction with one missing number. [closed]

Hi can some one please help me with the following subtraction? How is it done? $$\LARGE\frac{\genfrac{}{}{0pt}{0}{\hphantom{-}8617}{-39\hphantom{0} 9}}{\hphantom{-}8208}$$ Between $39$ and $9$ ...
1
vote
1answer
37 views

Need help factoring polynomial expression

I've started reading through a pre-calculus textbook for self-study and came across this problem in the second chapter: $$(x-2)^3-(x-2)^2$$ The final answer is $(x-2)^2(x-3)$ Everywhere I look but I ...
1
vote
0answers
36 views

Why zero to the zero power is 1? [duplicate]

The google calculator say that $0^0=1$. I'm confused. It's well-known $0^0$ is undefined.
1
vote
2answers
81 views

How to derive the formula for the sum of the first $n$ perfect squares? [duplicate]

How do you derive the formula for the sum of the series when $S_n = \sum_{j=1}^n j^2$? The relationship $(n, S_n)$ can be determined by a polynomial in $n$. You are supposed to use finite differences ...
0
votes
2answers
29 views

Proof with 3D vectors

Let ${a} = \begin{pmatrix}x_a\\y_a\\z_a\end{pmatrix}$, ${b} = \begin{pmatrix}x_b\\y_b\\z_b\end{pmatrix}$, and ${c} = \begin{pmatrix}x_c\\y_c\\z_c\end{pmatrix}$. Show that $(x_a,y_a,z_a)$, ...
-4
votes
4answers
75 views

Two girl's ages are $20$ when added and $99$ when multiplied. What is the age of each girl? [closed]

Two girl's ages are $20$ when added and $99$ when multiplied. What is the age of each girl?
-1
votes
2answers
50 views

What is the remainder when a polynomial $g(x^{12})$ is divided by $g(x)$? [closed]

Let $g(x) = x^5 +x^4 +x^3+x^2+x+1$. What is the remainder when the polynomial $g(x^{12})$ is divided by the polynomial $g(x)$?