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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votes
2answers
54 views

Largest range of a list of five integers in which the mean, median and mode are consecutive integers

A student notices that in a list of five integers, the mean, median and mode were consecutive integers in ascending order. What is the largest range possible for these five integers and why? Please ...
7
votes
5answers
110 views

How can I compare the numbers $2^{39}$, $5^{19}$ and $52^7$?

I have to compare the numbers $2^{39}$, $5^{19}$ and $52^7$. I don't know how to do that because their exponents don't have anything in common.
0
votes
0answers
54 views

How to know if there is a formula to find an answer?

I'm working on a problem and I've found all the values to make it work and I'm trying to figure out if there's a formula in there I can use. Here is what I have written out so far. ...
-1
votes
1answer
30 views

How to represent a $5$ digit number that has $62$ choices per digit?

If you have a $5$ digit number that can be 0-9A-Za-z how would you represent that? total_number_of_records = 5 digits * (10 + 26 + 26) ^ 5 I want to find out ...
2
votes
1answer
21 views

If A is the range of $f(x) = ^{7-x}C_x$ then the no. of reflexive relation from A to A is…

Problem : If A is the range of $f(x) = ^{7-x}C_x$ then the no. of reflexive relation from A to A is (a) $2^6$ (b) $2^{12}$ (c) $2^{16}$ (d)$2^{20}$ My approach : $f(x) = ^{7-x}C_x = ...
1
vote
4answers
51 views

Prove if $x > 100$ then $\frac{100}{3-2x} > -1$

If $x > 100$ then $\frac{100}{3-2x} > -1$ Pf: Assume $x>100$ $-2x > -20$ $-2x + 3 > -200$ $\frac{1}{3-2x} > \frac{1}{-200}$ This is what I have done so far, however I'm not ...
2
votes
2answers
30 views

Inverses of piecewise functions.

For an example, let $f: \mathbb{R}\rightarrow \mathbb{R}, $be defined by$ f(x) = 2x $ when x is rational and $f(x) = 3x$ when x is irrational. Can it simply be concluded that the inverse is ...
0
votes
4answers
44 views

How to prove b is positive? [closed]

If A * B is positive and I know that A is positive, how do I prove that B is positive? I have to prove it with just basic laws of numbers, e.g. associative property and stuff like that.
0
votes
2answers
41 views

Find all $z\in\mathbb{C}$ such that $e^z = 1$.

We write $z=a+ib$. Now, $$1 = e^z = e^{a+ib} = e^a e^{ib} = e^a(\cos b + i\sin b) = e^a\cos b + ie^a\sin b$$ We have $$1 = e^a \cos b \\ 0 = e^a\sin b$$ Now, I don't understand why it has to be ...
1
vote
2answers
15 views

Find $a$ and $b$ such that $a-b=c$ and the geometric mean of $a$ and $b$ is $m$.

Given $m,c\in\mathbb{R}^+$, how can I find numbers $a$ and $b$ such that $a-b=c$ and $m = e^\frac{\ln(a)+\ln(b)}{2}$ (i.e., $m$ is the geometric mean of $a$ and $b$). I understand that this doesn't ...
0
votes
3answers
72 views

Does differentiation of $f(x)=\log(x)$ yield two different results?

The two different results are :$\frac{1}{x}$ and $\frac{-1}{x}$. I read in my book that: $$\frac{d(\log x)}{dx}=\frac{1}{x}$$ where $x>0$ And: $$\frac{d(\log(-x)}{dx}=\frac{1}{x}$$ where ...
3
votes
2answers
40 views

How much did he spend the last $5$ day $?$

A man has a habit of spending an amount equal to the date on that day . For example Rs.$18$ on the $18$ th of the month and so on. Now on a fine day he is asked how much he had spent the last ...
1
vote
1answer
24 views

Limits w/ Greatest Integer and Abs. Value Function

Find the $$\lim _{x\to 2^+}\ {\lfloor x \rfloor - 1\over\lfloor x \rfloor - |x|}$$
3
votes
1answer
80 views

(Non-continuous) solutions to $f(f(x))=kx$ and $f(x^2)=xf(x)$

Given a fixed non-zero constant $k\in\mathbb{R}$, find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $$f(f(x))=kx\quad\text{and}\quad f\left(x^2\right)=xf(x).$$ If $f$ is continuous, ...
0
votes
3answers
34 views

Simplify $\sqrt[3]{36}*\sqrt[6]{\frac{4}{3}}*\sqrt{27}$ writing each factor in index notation

So I rearranged it in index form: $$36^{\frac{1}{3}}*27^{\frac{1}{2}}*({\frac{4}{3}})^{\frac{1}{6}}$$ After this i changed 36 into 4*9 and then 9 into $$3^2$$ So 36 became: ...
2
votes
1answer
55 views

How to calculate $4 \over {{x^4} + {y^4} + {z^4}}$ from $x + y + z = 1$ and other conditions more?

How to calculate $$4 \over {{x^4} + {y^4} + {z^4}}$$ from $$ x + y + z = 1, $$ $$ x^2 + y^2 + z^2 = 9, $$ $$ x^3 + y^3 + z^3 = 1. $$ Alternative answers: A) $1 \over {33}$, B) $2 \over {33}$, C) $4 ...
0
votes
1answer
27 views

Power series confusion when multiplying fractions.

I am stuck on the following question. check that the following sum from 0 to infinity converges using power series. sum of $$ 1/((n+(1/2))^2)$$ the next line of work is : $$4/((2n+1)^2)$$ I have ...
0
votes
3answers
53 views

Prove that: $2^n < n!$ Using Induction

I'm told to show that $2^n < n!$ using induction This is my attempt at it: BC: $n=4, 2^4 = 16 < 4!$ IH: n = k, $2^k < k!$ IS: try n = k+1 I'm told to only work from one side, so I try ...
1
vote
1answer
50 views

What is the largest value of $n$ where $\lg(n) \le 1,000,000$

What is the largest value of $n$ where $\lg(n) \le 1,000,000$ is the question that has been posed in a book I am currently working through and the answer is $2^{10^6}$. However I'm not sure how to get ...
0
votes
0answers
21 views

Express slope as a function of x

I'm stuck on this problem: Express the slope of the line joining $(1, 0)$ & any point on the graph of the semicircle below the x-axis, centered at the origin with radius one, as a function of ...
1
vote
2answers
26 views

Finding roots of cubic (trig)

The question is By putting $x$ $=$ $\frac 23 cos (\theta)$ Find the exact roots of the equation in terms of $\pi$ $$ 27x^3 - 9x = 1 $$ What I have attempted: $$ ...
2
votes
1answer
39 views

Prove: $ 1\times3 +2\times4 + \cdots + n(n+2) = \frac{1}{6} \times n(n+1)(2n+7)$ using Induction

I'm told to prove this by Mathematical Induction: $ 1\times3 +2\times4 + \cdots + n(n+2) = \frac{1}{6} \times n(n+1)(2n+7)$ This is what I have so far: BC: Try $n=1$: $ 1\times3 +2\times4 + \cdots ...
0
votes
2answers
37 views

Absolute Value Rational Inequalities

Ok so I have the following two inequalities: \begin{equation} \left| \frac{x+6}{x-2}\right| \leq 4 \end{equation} and \begin{equation} \frac{x^2-1}{\left| x+2\right|} \leq 3(1-x) ...
0
votes
2answers
51 views

If $m = 6x + 5$, what equation is equivalent to $(6x + 5)^2 - 10=-18x - 15$ in terms of $m$?

Hey guys at Mathematics Stack Exchange, I have a question for you guys. This question comes from the Khan Academy practice task entitled 'Solve quadratic equations by using structure', so the credit ...
1
vote
2answers
47 views

Equation with radicals and reciprocals

Find all $x\in\mathbb{R}$ satisfying $$x=\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}.$$ Multiply both sides by $x^{1/2}$ to get $$x^{3/2} = \sqrt{x^2-1} + \sqrt{x-1}.$$Making the substitution $a = ...
1
vote
1answer
43 views

Find integers $x$ and $y$ such that $\frac{27^{x+y}}{9^{xy}}=27$ and $\frac{4^{2xy}}{8^{x+y}}=512$ .

Find all the integers $x$ and $y$ such that : $$\frac{27^{x+y}}{9^{xy}}=27$$ and :$$\frac{4^{2xy}}{8^{x+y}}=512$$ I'm in Algebra two and I feel like there are certain types of math I haven't ...
0
votes
1answer
33 views

Expanding an expression in a certain field

If $\mathbb F_2$ is a field of characteristic $2$, then we have $x+x=y+y=z+z=0$ for all $x,y,z \in \mathbb F_2$. When I expand $(x+y)(y+z)(z+x)$, I get \begin{align} (x+y)(y+z)(z+x) &= ...
-1
votes
2answers
56 views

Odd and Even Numbers [closed]

New Info I didn't understand the concept of odd and even numbers hence I asked the question. No need for downvote or hold :-( Thanks to all who helped! Original Question GRE Exam Guide asks If ...
0
votes
2answers
36 views

How to substitute correctly in this equation?

So I have this equation: max_rep = (kg * rep * 0.0333) + kg By providing the value of max_rep and ...
0
votes
1answer
19 views

Find the lenght of a rectangle between two parabolas

I'm trying to find the length of $PQ$ but the best thing I have done so far is finding that the point $T$ is $(0,4)$, as well as finding the distance between the two turning points to be $6$. Can ...
1
vote
0answers
34 views

Prove the identity $\tanh(N\textrm{acosh}\;a) = \vert \frac{g^{2N}-1}{g^{2N}+1}\vert$

During my recent study, I found an Identity which is of the form $$ \tanh(N\textrm{acosh}\;a) = \left\vert \frac{g^{2N}-1}{g^{2N}+1}\right\vert $$ where $a\geq1$ and $g>0$ satisfy ...
1
vote
1answer
43 views

How sin(90°+ θ) is equal to M'P'/OP' or Cos θ?

I'm learning Trigonometry right now with myself and at current I'm understanding how to find the trigonometric ratio of the angle (90°+ θ) in those of θ. I'm little bit confused right now in the ...
-1
votes
3answers
49 views

$\sin A \in [-1,1]$ and $\cos A \in [-1,1]$. Then why is $\tan A =$ more than $1$ or less than $-1$ [closed]

$\sin A \in [-1,1]$ and $\cos A \in [-1,1]$. Then why is $\tan A $ outside of $[-1,1]$?
0
votes
0answers
22 views

Irrationality of a rapidly converging function.

Let $$\exp{\int_{2}^{\infty} \frac{f(x)}{x^9 - x}\,\mathrm d{x}} = K$$ where $\lim_{x \to \infty} \frac{f(x)}{x} = 0$. Is $K$ irrational for any such $f(x)$ where $x\geq 2$ is an integer? My ...
2
votes
3answers
49 views

How to show that $\frac{\ln x}{x}$ is monotone for $x\ge e$?

How to show that $\frac{\ln x}{x}$ is monotone for $x\ge e$? Looking at the graph of $\ln x$ I can tell that for $x<e$ the $\ln x$ goes to $-\infty$ very fast and for $x\ge e$ it grows very slow. ...
3
votes
1answer
39 views

What is the range of $y$ if $x+y+z=4$ and $xy+yz+xz=5$ for $x, y, z \in\mathbb{R}_+$

What is the range of $y$ if $x+y+z=4$ and $xy+yz+xz=5$ for $x, y, z \in\mathbb{R}_+$ How to explain the following method? Let $x=z$ then: $$2x+y=4\quad;\quad 2xy+x^{2}=5$$ $$\implies \left( ...
0
votes
1answer
53 views

If $f(x-2)=x$ for all real numbers x, then what is $f(x)$?

If $f(x-2)=x$ for all real numbers x, then $f(x)=?$ I think the answer stays the same, because the given says for all real x. so is $f(x)=x$ or i am wrong?
2
votes
2answers
28 views

Quartic with $4 $ equidistant roots

Today I got the problem $(x^2 -1)(x^2 -4)=k$, and I have no idea how to prove this algebraically. $K$ is a real, non-zero number that makes the equation have $4$ distinct real equidistant roots. Solve ...
2
votes
3answers
75 views

Finding $(a+\sqrt b)^n+(a-\sqrt b)^n$ where $n$ is natural

For the expression $\left(a+\sqrt{b}\right)^n+\left(a-\sqrt{b}\right)^n$ where $n \in \mathbb{N}$, and $a,b, \in \mathbb{Q}$, the radical is always ends up cancelled, and the result is always in ...
3
votes
1answer
44 views

A smart way to do this question.

Let $S=\{0,1,2,\dotsc,25\}$ And $T=\{n\in S : n^2+3n+2\text{ is divisible by }6\}$ Then the number of elements in $T$ is? One way I know is to factorise it as $(n+1)(n+2)$. And then put each $n$ ...
0
votes
2answers
39 views

How do I simplify this expression involving the exponential function

$ \frac{e^{16}-e^{-16}}{e^8-e^{-8}}$ How do I simplify this expression involving e to $\frac {1+e^{16}}{e^8}$? I have tried multiplication and division by e^8 and e^16 and simplifying the ...
4
votes
1answer
53 views

Find the equation of parabola passing through $(-1, 6), (1, 4), (2, 9)$

A(the?) equation of parabola is $y = ax^2 + bx + c$. That gives the equations below: \begin{align*} 6 & = a - b + c\\ 4 & = a + b + c\\ 9 & = 4a + 2b + c \end{align*} Then I simply ...
0
votes
1answer
48 views

How many possible values are there for $k$?

If $6x+1.5y+0\cdot z = k$ and $x+y+z = 25$ where $x,y,z$ are nonnegative integers, how many possible values are there for $k$? I would solve this by substituting in for $x$ to get ...
0
votes
1answer
38 views

Confusion with algebraic manipulation

$$\left(\frac{5}{5}\right)^n \cdot \left(\frac{9}{10}\right)^n = \left(\frac{1}{5}\right)^n \cdot \left(5 \cdot \frac{9}{10}\right)^n $$ I would like for someone to explain in laymen's terms how to ...
0
votes
3answers
90 views

Proof that $(3\cdot 2^n-1)$ is not a multiple of $17$ for any value of $n$ [closed]

Prove that $3\cdot 2^n-1$ is not a multiple of $17$ for any positive integer $n$.
0
votes
0answers
36 views

Any set of 3 linearly independent vectors is a basis in $R^3$

In other words, given any $\vec{v} = (v_1, v_2, v_3)$ and a linearly independent set of vectors $\{\vec{x},\vec{y},\vec{z}\}$, there exists $\alpha, \beta, \gamma$ such that $\alpha\vec{x} + ...
5
votes
1answer
70 views

Summation of factorials.

How do I go about summing this : $$\sum_{r=1}^{n}r\cdot (r+1)!$$ I know how to sum up $r\cdot r!$ But I am not able to do a similar thing with this.
-1
votes
3answers
72 views

How to approach solving for $x$? $(1+x)^2 = 1.21$

I'm trying to solve for $x$ in the following equation, but don't know how to solve it without using a graphing calculator. $$(1+x)^2 = 1.21$$ Is there a rule I'm supposed to follow?
2
votes
1answer
36 views

Simplifying hyperbolic compositions like $\sinh (N \operatorname{acosh} a)$

In many occasions, we may meet hyperbolic functions, as well as their combined ones. I want to simplify expressions like $$ \tanh\left( N\left(\textrm{acosh}~ a\right)\right) $$ and $$ \sinh\left( ...
0
votes
0answers
21 views

What is the form $ca^{p}b^{q}$?

What is the form $ca^p b^q?$ I am to write this expression $$\left(\frac{a^{2/3}}{b^{1/2}}\right)^2 \cdot \frac{b^{3/2}}{a^{1/2}}$$ in the $ca^p b^q$ form, but, I've NEVER heard of it before today. ...