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
65 views

Find the range of values for k such that ${kx^2 + 3x + 9k = 0}$ has real roots

I am asked the question: Find the range of values for ${k}$ such that ${kx^2 + 3x + 9k = 0}$ has real roots. So from my understanding, there are distinct roots if ${b^2 - 4ac\ge 0}$ My first step ...
2
votes
2answers
34 views

Functions and Graphs assumption of same algebra equation.

$$f(x) =\frac{ (x - 1) }{(x^2 - 1)}$$ $$g(x) = \frac{1}{(x + 1)}$$ Is $f = g$? why? Solution: answer is no. I don't get it. why can't it be same when it is the alt form?? Can anyone explain this ...
0
votes
1answer
46 views

Do we have $\frac{1}{a} - \frac{1}{b} = b - a$?

I am attempting to prove that $$\frac{1}{E'} - \frac{1}{E} = \frac{1}{m_e c^2} \cdot (1-\cos\theta)$$ can be derived from $$E + m_ec^2 - E' = c^2(p^2 - 2pp'\cos\theta + p'^2) + m_e^2c^4 $$ where ...
0
votes
4answers
36 views

Simple Fraction needing explanation

$$\frac{x}{x^{-1/2}} = x^{3/2}$$ How? I don't see what is going on here. What rule is being used to achieve this amount?
0
votes
1answer
18 views

How to use Rolle's theorem to verify the following?

How to use Rolle's theorem to verify the location of roots ? $f(x)=x^3+4/x^2+7$ has exactly one zero in ($-\infty$,$0$) I can do it without Rolle's theorem by finding the stationary point which is ...
0
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1answer
14 views

Difficulty simplifying nested sums with different variables

I'm trying to work out an algorithm analysis problem, and I'm having some difficulty determining how a jump is made between two steps in the answer. $$ \begin{align} ...
3
votes
4answers
41 views

$2x(1-x)$ is not onto?

How come $4x(1-x)$ is onto in $[0,1]$ but $2x(1-x)$ is not? Isn't it true that for any $y$ in the range interval, there exist two $x$ such that $f(x)=y$?
0
votes
2answers
17 views

Find real points of intersection of the equations algebraically [on hold]

Can you help me to find pints of intersection of given system of equesions? Can't do it by myself. $xy + x - 2y + 3 = 0$ $x^2 + 4y^2 - 9 = 0$
42
votes
1answer
1k views

Find the limit $L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$

Find the limit following: $$L=\lim_{ _{\Large {n\to \infty}}}\:\sqrt{\frac{1}{2}+\sqrt[\Large 3]{\frac{1}{3}+\cdots+\sqrt[\Large n]{\frac{1}{n}}}}$$ P.S I tried to find the value of $\:L$, but I ...
0
votes
1answer
77 views

How to solve this Partial Decomposition fraction?

I was doing my laplace transform revision and I came across this problem. How to decompose this partial fraction ? $$ \frac{145s}{(s^2+4)(s^2+4s+13)} +\frac{9s+55}{s^2+4s+13}$$ How to write the ...
0
votes
3answers
31 views

Average rate of change at an exact point

So I had this question in my precalculus text book: Water is draining from a large tank. After $t$ minutes there are $160,000-8000t+t^2$ gallons of water in the tank. Estimate the rate at which ...
1
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3answers
20 views

simplifying complex fractions when proving inverses with function composition?

I'm working with two functions, $f(x)=\frac{x-3}{x+4}$ and $g(x)=\frac{4x+3}{1-x}$. I need to simplify $f(g(x))$ and its opposite, but i'm not sure of the procedures regarding the more complicated ...
0
votes
0answers
22 views

Range of inverse harmonic mean of two integers

Today I was solving an exercise and one of the things I tried (which later turned out to be useless) involved considering the following: Is there a simple way to describe in terms of $n$ the range of ...
2
votes
4answers
74 views

If $x^2+3x+5=0$ and $ax^2+bx+c=0$ have a common root and $a,b,c\in \mathbb{N}$, find the minimum value of $a+b+c$

If $x^2+3x+5=0$ and $ax^2+bx+c=0$ have a common root and $a,b,c\in \mathbb{N}$, find the minimum value of $a+b+c$ Using the condition for common root, $$(3c-5b)(b-3a)=(c-5a)^2$$ ...
2
votes
4answers
45 views

Show that $6^n/n! \le 6^5/5! \times 6/n$

I want to show that $$\frac{6^n}{n!} \le \frac{6^5}{5!} \cdot \frac 6n$$ without using induction, which I've done but is rather clunky. Is there a more straight forward way of doing this?
1
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0answers
14 views

Find the rule connecting $y$ and $x$ in the form $y=mx+c$ for $1<x≤2$ [on hold]

I have attempted numerous methods but I'm not sure how to interpret the $1<x≤2$ into a rule relating $y$ and $x$.
4
votes
3answers
32 views

Prove that $a(x+y+z) = x(a+b+c)$

If $(a^2+b^2 +c^2)(x^2+y^2 +z^2) = (ax+by+cz)^2$ Then prove that $a(x+y+z) = x(a+b+c)$ I did expansion on both sides and got: $a^2y^2+a^2z^2+b^2x^2+b^2z^2+c^2x^2+c^2y^2=2(abxy+bcyz+cazx) $ but ...
3
votes
5answers
63 views

Why is the solution to $\sqrt{6-5x}=x$ only $x=1$ and not $x=-6$? [duplicate]

I solved the equation $\sqrt{6-5x}=x$ as follows: $$(\sqrt{6-5x})^2=x^2$$ $$6-5x=x^2$$ $$0=x^2+5x-6=(x+6)(x-1)$$ $$x=-6 \quad \text{or} \quad x=1$$ If I plug in $x=-6$ into the original equation, I ...
0
votes
5answers
3k views

Finding two numbers when having their sum and product

I have two numbers, their sum is 41 and their product is 238. What are the numbers? I got during this far in my calculations: $a+b=41,\quad ab=238,\quad 238=41-b.$ I appreciate answers or tips to ...
-2
votes
2answers
76 views

Solve for $X: X^{49} = 60$

An unknown value is raised to a known power which results in another value, also known. How do you find the unknown value?
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3answers
71 views

Distance between a circle and a line

Find the distance between the circle $(x-3)^2+(y+2)^2=4$ and the line $x + 2y = 9$.
2
votes
2answers
102 views

Mathematics Olympiad Question $a+b+c=7$, …

Given $a+b+c=7$ and $\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a} = 0.7$, need to find $\frac{c}{a+b} + \frac{a}{b+c} + \frac{b}{a+c}$. I have noted that these two differ by a factor of $10$. So I ...
4
votes
1answer
52 views

Polynomial divides set of points

Given a set of points in the plane with distinct $x$-coordinates, each point colored black or white. A polynomial $P(x)$ "divides" the set of points if no black point lies above $P(x)$ and no white ...
0
votes
2answers
40 views

Prove $(a_1 + a_2 + a_3)^2 = a_1^2 + a_2^2 + a_3^2 + 2(a_1a_2+ a_2a_3 + a_1a_3)$

I wish to find a proof for this equality: $(a_1 + a_2 + a_3)^2 = a_1^2 + a_2^2 + a_3^2 + 2(a_1a_2+ a_2a_3 + a_1a_3)$ But then I realized there exists a more general version: $(\sum\limits_{k=1}^n ...
0
votes
1answer
369 views

A family of functions graphing

(1.) A family of functions is given. Graph all the given members of the family in the viewing rectangles indicated. f(x) = (x − c)^2 $$c = 1, -1, 3, -3; [−5, 5] \times [−10, 10]$$ (2.) A family of ...
0
votes
1answer
25 views

Tangent meets curve again

If the tangent at the point $(16,64)$ on the curve $y^2=x^3$ meets the curve again at at $Q(u,v)$ then $uv$ is ? If found the tangent to the curve at $(16,64)$ but then I cannot find $uv$.Give your ...
0
votes
1answer
40 views

The uniqueness of roots of Quartic function

Define $$ f(x):=(1+ax)^3x-a(1+x)^3. $$ Would it be possible to prove that the function $f$ has only one positive real root provided that $a>0$? (There might be another root $x_0<0$, but I only ...
2
votes
1answer
71 views

How does one prove that $2\uparrow\uparrow16+1$ is composite?

Just to be clear, close observation will show that this is not the Fermat numbers. I was reading some things (link) when I came across the footnote on page 21, which states the following: ...
0
votes
2answers
27 views

How to rewrite an expression

Let's say $Z=Y_1+Y_2$. I have this expression: $Y_1!Y_2!$. I want to rewrite the expression and express it by only $Z$. Is that possible?
4
votes
2answers
76 views

Are there any other solutions to this equation?

Consider the equation $1-t = tx^{1-2t}$ for some complex number $t$ and real $x$. Are there any other solutions to this equation besides $\Re(t) = \frac{1}{2}$ ? My attempt: The above equation can be ...
1
vote
2answers
42 views

Does the inverse of $f(x)=x^3$ have a non-negative domain to have a real output?

I'm not familiar with complex analysis. While playing with Mathematica (a mathematics software), I found that it keeps spitting out unexpected results, and the reason was that it considers differently ...
1
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0answers
30 views

Use the arithmetic-geometric inequality for this list to deduce the arithmetic-geometric inequality for $n$.

Suppose that $n$ is not a power of two. Let $2^k$ be a power of $2$ that exceeds $n$ and consider the list $$a_1,\dots,a_n,\underbrace{A,A,\dots,A}_\text{$2^k-n$ times}$$ of length $2^k$. Use the ...
0
votes
1answer
23 views

Line of intersection between two planes

The questions asks: Determine the line of intersection in vector, parametric, and Cartesian between the following sets of planes: $2x-y+2z+1=0$ and $-4x+2y-4z-2=0$ I realize these are parallel to ...
-3
votes
1answer
32 views

Formula $\sqrt[x]a -\sqrt[x] b$ [closed]

It is correct ? 1.) $$(\sqrt[x]{a} - \sqrt[x]{b}) \dot\ \sum_{k=0}^{\ x-1} a^{\frac{\ x-k-1}{x}} \dot\ b^{\frac{\ k}{x}}=a-b$$ 2.) $$ \lim\limits_{x \to \infty} \sum_{k=0}^{\ x-1} a^{\frac{\ ...
1
vote
1answer
18 views

Creating a four dimensional system to find a point

How can I create a four-dimensional system that has a solution of $(-2,5,-6,1)$? I know how to solve a system for its solution, but how do I work backwards?
4
votes
2answers
71 views

Solve $\sqrt[3]{7x+19}+\sqrt[3]{7x-19}=\sqrt[3]{2}$ by algebraic methods

I was trying to solve this equation without using calculus. Is it possible to be solved by elementary algebraic methods? $$\sqrt[3]{7x+19}+\sqrt[3]{7x-19}=\sqrt[3]{2}$$
0
votes
3answers
46 views

How to prove geometrically the difference between line and plane?

Explain geometrically why $(x,y,z) = (1,9,17) + s(1,1,1) + t(-2,-2,-2)$ represents the equation of a line and not a plane. EDIT: the extension to this question asks "based on the above answer, does ...
3
votes
2answers
47 views

Representation of roots of unity.

How to represent solutions of $\sqrt[26]{1}$ with solutions of $\sqrt[26]{-1}$? I know that $$w_{k}=\cos\left(\frac{0+2k\pi}{26}\right)+i\sin\left(\frac{0+2k\pi}{26}\right), \; \; ...
1
vote
2answers
51 views

For which $a$ and $b$ the function $\frac{ax+b}{a^2x+b^2}$ is increasing?

For which $a$ and $b$ the function $$\frac{ax+b}{a^2x+b^2}$$ is increasing? I know that function is increasing if $x_1 > x_2 \implies f(x_1)>f(x_2)$ but how can I find $a$ and $b$ for ...
-1
votes
1answer
16 views

Write an equation for a rational function with:

Write an equation for a rational function with: Vertical asymptotes at $x = -5$ and $x = 5$ $x$ intercepts at $x = -2$ and $x = -6$ Horizontal asymptote at $y = 6$ $y =$ ? I have ...
2
votes
0answers
45 views

If $a+b=8$ and $ab+c+d = 23$ and $ad+bc=28$ and $cd=12\;,$ Then $abcd$

If $a+b=8$ and $ab+c+d = 23$ and $ad+bc=28$ and $cd=12\;,$ Then value of $(1)\;\; a+b+c+d=$ $(2)\;\; ab+bc+cd+da = $ $(3)\;\; abcd=$ My attempt: Let $x=a\;,b$ be the roots of ...
3
votes
3answers
38 views

Is $f(x)=x^{3}+3x^{2}+12x-2\sin x $ one-one and onto?

For linear or simple quadratic equations, it is quite simple to check if the function is onto or not. But I often face questions like the one I posted above, to check whether they are one-one and ...
0
votes
2answers
163 views

What number goes into both 225 and 100?

I'm doing a bit of algebra/pre-calculus. What would be the highest number that goes into both $225$ and $100$?
2
votes
0answers
9 views

Vector equation of line containing point and perpendicular to plane [duplicate]

How would one find the vector equation of the line that contains the point (x0, y0, z0) and is perpendicular to the plane Ax + By + Cz = D?
1
vote
1answer
75 views

Square root branch cut

Consider the following expression: \begin{equation} \phi(\delta)=i\,\sqrt{-3+i\,\delta}, \end{equation} where $\delta$ is infinitesimal. If we choose a branch cut along the negative real axes, it ...
1
vote
1answer
36 views

How to find $f(2)+f^{-1}(5)$ if $f(2x^2+3x+4)=6x^2+9x+20$? [closed]

$$f(2x^2+3x+4)=6x^2+9x+20$$ How to solve $f(2)+f^{-1}(5)$ ? Any help or advice on solving is much appreciated. Thanks!
0
votes
2answers
36 views

Finding x-intercept of a parabola given one x-intercept

I am given an $x$-intercept of $-3-\sqrt{7}$ and I am asked to find the other intercept. I am having trouble since I don't have any other information but the given $x$-intercept. My guess is that the ...
0
votes
1answer
30 views

Let $f(x)=p\cos x+q\sin x,|p|+|q|\ne0$ and $|f(x)|\leq 1$.Let $\alpha,\beta$ be the roots of the equation $f(x)=1,|\alpha-\beta|=k\pi,k\in R,$

Let $f(x)=p\cos x+q\sin x,|p|+|q|\ne0$ where $p,q\in R$ and $|f(x)|\leq 1$.Let $\alpha,\beta$ be the roots of the equation $f(x)=1,|\alpha-\beta|=k\pi,k\in R,$then the find the possible values of $k.$ ...
2
votes
2answers
90 views

Proving $x>\sin(x)$ without calculus for $x>0$

The starting problem was to prove $$\sin 26^{\circ}\sin 58^{\circ}\sin 74^{\circ}\sin 82^{\circ}\sin 86^{\circ}\sin 88^{\circ} \sin 89^{\circ}>\frac{45\sqrt{2}}{64\pi}\\\cos 1^{\circ}\cos ...
0
votes
1answer
72 views

Solution to the equation $x^3-3=2\sqrt{x+2}$

Solve the equation $x^3-3=2\sqrt{x+2}$. I have tried to let $t=\sqrt{x+2}$ then we have $$\begin{cases} x^3-3&=2t \tag 1\\ t^2 &=x+2 \end{cases}$$ But I've stuck here... Any help ...