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
42 views
-1
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4answers
57 views

Solving for Roots/Zeros [closed]

$$0=\frac{x}{4\sqrt{x^2+16}} - \frac{1}{10}$$ How would i go about solving this? I have started by bringing the -1/10 to the other side but after that I'm stuck and don't know what to do...
0
votes
0answers
35 views

If $f(-x-f(0))=f(x)+f^2(0)+2xf(0)$ is it possible to find $f(0)$

Given the functional equation $f(-x-f(0))=f(x)+f^2(0)+2xf(0)$ if we substitute $x=0$ we get $f(0)=f(-f(0))-f^2(0)$ but if we substitute $x=f(0)$ we get $f(-f(0))=f^2(0)+f(0)$, which is the same ...
0
votes
1answer
28 views

Solving a cubic function with P and Q

I have been struggling a little bit over solving cubic functions. I have been trying to use the P and Q method. So the question is What is the approximate value of the greatest zero of $f(x) = x^3 - ...
0
votes
2answers
30 views

Reducing algebraic fractions

$$\frac{ 9,009x^{4/3}y^2 - 7,007x^{7/3}y }{ 4,004x^{1/3}y }$$ $$\frac{ 1,001x^{4/3}y (9y - 7x^1) }{ 4(1,001)x^{1/3}y }$$ $$\frac{ x(9y - 7x) }{ 4 }$$ How is it ${ 7x }$ it looks like it should be ...
1
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2answers
37 views

How to find the quotient of a rational expression

I have been stuck on this problem because I don't know how to find the quotient of a rational expression. $$\frac{x^4 - 1}{x + 1}$$ Thanks
3
votes
2answers
50 views

If $f(-f(x))=f(-f(-x))$ can we conclude that $f(x)=f(-x)$?

If $f(-f(x))=f(-f(-x)), \quad f:\mathbb{R}\rightarrow\mathbb{R}$ can we conclude that $f(x)=f(-x)$? It seems unlikely, but I'am trying to solve a functional equation where the solution seems to be ...
1
vote
1answer
65 views

How do I show that $\frac {\cos^2 A}{\cos^2 B} + \frac {\cos^2 B}{\cos^2 C} + \frac {\cos^2 C}{\cos^2 A} \ge 4(\cos^2 A + \cos^2 B + \cos^2 C)$?

Let $A, B, C$ be the angles of an acute triangle. Show that $$\frac {\cos^2 A}{\cos^2 B} + \frac {\cos^2 B}{\cos^2 C} + \frac {\cos^2 C}{\cos^2 A} \ge 4(\cos^2 A + \cos^2 B + \cos^2 C).$$ How should ...
2
votes
4answers
83 views

How do I show that $\sum_{i = 1}^n \frac 1{\sqrt{a_n}} \lt \frac {\sqrt 3}6$ for $a_n = 4n(4n + 1)(4n + 2)$?

Let $a_n = 4n(4n + 1)(4n + 2)$, show that $$\sum_{i = 1}^n \frac 1{\sqrt{a_i}} \lt \frac {\sqrt 3}6 \quad \forall n \in \mathbb{N}^+.$$ I know I need to find an upper bound for $1/\sqrt{a_n}$ but I ...
1
vote
1answer
41 views

How is the following proof really a proof (inequality)?

The user has just subtracted by $\frac{a}{b}$ in the first step and then rearranged the terms to show that it's positive and similar steps have been used to prove the second part of inequality. How ...
0
votes
2answers
76 views

What's wrong with my solution of inequality?

Question: solve the following inequality: $\frac{x}{2} \geq \frac{5}{x + 1} + 4$ My solution: $\frac{x}{2} - \frac{5}{x + 1} + 4 \geq 0$ $\implies \frac{x(x + 1) - 10(x + 1) - 8(x + 1)}{(x + 1) 2} ...
0
votes
1answer
27 views

When should one use a closed interval and when an open one in inequality?

In the following solution: In case I, the person has taken $2x \geq 0$ and then solved the equation. For the other inequality, he has taken $3 - x \gt 0$ and then solved the equation. My question ...
1
vote
2answers
24 views

Let $A\subset \mathbb{R}$ such that $l=\text{inf }(A)$ exists. Prove that $\forall \epsilon >0 $ there is $a\in A$ in the interval $[l,l+\epsilon)$

I need to prove the following: Let $A\subset \mathbb{R}$ such that $l=\text{inf}(A)$ exists. Prove that $\forall \epsilon >0 $ there is $a\in A$ in the interval $[l,l+\epsilon)$ That's what I ...
0
votes
3answers
70 views

Proving $\frac{n^2}{n-3}$ diverges

I need to prove that $$\frac{n^2}{n-3}$$ diverges For that, I need to prove that, given $\epsilon>0$, we have $n_0$ which depends on $\epsilon$, such that: $$n>n_0 \implies ...
3
votes
4answers
95 views

Real solution of the equation $\sqrt{a+\sqrt{a-x}} = x\;,$ If $a>0$

For a real number $a>0\;,$ How many real solution of the equation $\sqrt{a+\sqrt{a-x}} = x$ $\bf{My\; Try::}$ We can Write $\sqrt{a+\sqrt{a-x}} = x$ as $a+\sqrt{a-x}=x^2$ So we get ...
1
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3answers
41 views

Stirling on ${2n-1 \choose n}$

I'm trying to find an expression for $${2n-1 \choose n}$$ using Stirling's approximation $$k!\sim \sqrt{2\pi k}(\frac{k}{e})^k.$$ I see $${2n-1 \choose n}\approx ...
2
votes
0answers
112 views

Is $\dfrac{\cos\theta}{\sqrt{15}}$ irrational? [closed]

In general I was wondering if $\cos\theta$ was between $0$ and $1$ exclusive then would $\dfrac{\cos\theta}{\sqrt{15}}$ be irrational? And just on another note is an irrational times a transcendental ...
0
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0answers
34 views

rational function cancellation [duplicate]

This is probably a trivial question however i cannot find the correct information online. When simplifying mappings from $\mathbb{R}$ to $\mathbb{R}$ such as: $$\frac{x(x-1)}{(x-1)}$$ Why is it ...
1
vote
1answer
59 views

how do I solve $y -\sin y= 1$

I am trying to use trigonometric equations to solve $y - \sin y = 1$, such as solving for $y$ but it is not working out, I have found $\cos y = \sqrt{-y^2 + y}$ but it does not lead to anywhere ...
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5answers
50 views

$x<y \iff x^{-1}>y^{-1}$

In order to prove the following: $$x<y \iff x^{-1}>y^{-1}$$ *for $x>0$ and $y>0$ I tried this: $$x<y\implies y-x>0$$ I have to prove that this, implies that ...
0
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1answer
38 views

Showing a function is odd

I have this equation: $$ f(x) = \frac{2x^2+3}{x-2} $$ and I have to prove it has half-turn symmetry around the point (2,8). I know that for a function to have half-turn symmetry, it needs to have ...
1
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3answers
31 views

Solve for a in Exponential equation

Is it possible to solve for $a$ in the following equation: $a^\alpha=b^\alpha-a$? Currently, I have resorted to using Excel to approximate $a$ (I am given values for $b$ and $\alpha$), but am ...
0
votes
1answer
32 views

Can someone show me the steps to simplify this?

$-x+x^{1.5}-2x^{1.5}+2x^{2}$ $=$ $2x^2-x^{1.5}-x$ I'm having trouble simplifying this. Can someone show me the steps?
0
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0answers
31 views

How useful is Prentice Hall Algebra 2?

I am currently a sophmore in highschool, and I wish to continue learning mathematics over the summer. My school laptop has a copy of Algebra 2 published by Prentice Hall. I have so far been unable to ...
1
vote
1answer
24 views

For what maximum positive $k$ is $2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$ true?

I am trying to find the maximum value of $k$ such that the inequality $$2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$$ is satisfied. I impose restrictions that $n \in \mathbb{Z}$ with $n \geq ...
1
vote
1answer
26 views

Finding parametric equations of rectangular equation

Is there a general process to follow when finding the parametric equations of a normal rectangular equation ? I know that one rectangular equation might have many parametric equations, but are there ...
0
votes
3answers
22 views

How can we find out the interval in an inequality?

Please go through the following link: Why is equating one of the bracks to zero in this equation correct? Now, the expression given there is $(x+1)(x+3)$, I understand now why we take either of these ...
0
votes
1answer
18 views

Explanation of homogenous function

Is there someone, who can explain why the function $g(s)=f(e^s,e^s)$ is not homogeneous when it can be written as $\frac{9}{4}e^{s/2}s$. I got the function $f(x,y)=\sqrt x +2\sqrt y +\frac{3y}{\sqrt ...
2
votes
1answer
136 views

A problem of olympiad. [closed]

This nice functional equation was proposed in the “VIII Olimpíada Iberoamericana de Matemáticas” held in Mexico (1993). Find all the functions $f:\mathbb N^* \to \mathbb N^*$ such that i) ...
2
votes
2answers
35 views

Squeeze Theorem with a restriction

Hi there I have this question: $$4x - 9 \leq f(x) \leq x^2 - 4x + 7$$ for $x \geq 0.$ Find the limit of $f(x)$ as $x$ approaches $4$ I know the answer is seven but why does the question inform me ...
1
vote
1answer
18 views

Construct sinusoidal functions

Can anyone explain how do I determine if the amplitude is positive or negative? I dont quite understand the explanation given here.
4
votes
3answers
119 views

How do you prove this without using induction?

How do you prove this without using induction $$\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n-1}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1}{2n-1}$$
0
votes
1answer
26 views

Can you raise a number to the power of another number being raised to a power?

So I actually have two questions. Is it even Possible to raise a number to the power of a number with its own exponent? Kind of like an exponent within an exponent....? It doesn't sound right to me ...
0
votes
1answer
67 views

Trigonometric equation: $\sin^3x+\cos^3x+\sin^2x+\sin x+\cos x=2$.

Well, the title says it all. I've tried utilizing the fact that $\sin^3x+\cos^3x=(\sin x+\cos x)(1-\sin x\cos x)$ and then the equation becomes $(\sin x+\cos x)(2-\sin x\cos x)=1+\cos^2x$. Squaring ...
2
votes
1answer
29 views

Projectile motion: Proving:$ x^2 + 4 \left(y-\frac{v^2}{4g} \right)^2 = \frac{v^2}{4g^2} $

Question: Projectiles are fired with initial speed $v$ and variable launch angle $0< \alpha < \pi$. Choose a coordinate system with the firing position at the origin. For each ...
1
vote
5answers
90 views

How to evaluate $\arctan(1)$

I thought $$\arctan(1) = \dfrac{\arcsin(1)}{\arccos(1)}$$ Sin hits 1 at $\pi/2$, and Cos hits 1 at $0$ and $2\pi$ So $\dfrac{\arcsin(1)}{\arccos(1)} = \dfrac{1}{4}$ But the solution says it is ...
3
votes
2answers
65 views

Find the value of $ab+ 2cb+\sqrt3 ac$?

Three positive real numbers $a,b,c$ satisfy the equations $a^2+\sqrt3 ab+b^2=25$, $b^2+c^2=9$ and $a^2+ac+c^2=16$ .Then find the value of $ab+ 2cb+\sqrt3 ac$? Is there some way to find the desired ...
1
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2answers
41 views

How to solve $\frac{1}{(|x| - 3)}$ $\lt$ $\frac{1}{2}$?

$\frac{1}{(|x| - 3)}$ $\lt$ $\frac{1}{2}$ $x$ can be $\ge$ 0 or $\le$ 0 Case 1 :- $x$ $\ge$ 0 $therefore$, $\frac{1}{(x - 3)}$ $\lt$ $\frac{1}{2}$ $\Rightarrow$ $\frac{1}{(x - 3)}$ - ...
1
vote
1answer
20 views

Is this solution for a broadwalk problem correct?

Question: Distance, Speed, and Time A boardwalk is parallel to and 210 ft inland from a straight shoreline. A sandy beach lies between the boardwalk and the shoreline. A man is standing on ...
1
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0answers
46 views

Defining Logic Algebraically, Math Functions & Integers

Introduction I wanted to define some functions algebraically to be used as "logical conditions" that would be assigned to a term $t$ to "control" its value. Or in some other words, I wanted to ...
2
votes
1answer
66 views

Solve given equation $4^{(x-2)(x+3)} - 64^{(x-3)} = 0?$

Solve given equation $4^{(x-2)(x+3)} - 64^{(x-3)} = 0?$ My attempt: I've attempted to solve this question, but isn't it impossible to solve, i.e has already been simplified completely? ...
3
votes
2answers
53 views

Finding Roots of tenth degree polynomial

I know that there are no explicit formulas to find roots for polynomials of degree higher than $4$. I have to find all the roots of the polynomial $ f(z) = 1+z^2+z^4+z^6+z^8+z^{10}$ I found two ...
0
votes
2answers
63 views

Wolfram answer is different for the integral $\sqrt{\frac{x}{2-x}}dx$

$$I=\sqrt{\frac{x}{2-x}}dx=\int \frac{xdx}{\sqrt{2x-x^2}}=\frac{-1}{2} \times \int\frac{(2-2x-2)dx}{\sqrt{2x-x^2}}$$ so $$I=\frac{-1}{2}\int\frac{(2-2x)dx}{\sqrt{2x-x^2}}+\int ...
3
votes
3answers
83 views

Integrating $\int \frac{\sqrt{x^2-x+1}}{x^2}dx$

Evaluate $$I=\int\frac{\sqrt{x^2-x+1}}{x^2}dx$$ I first Rationalized the numerator and got as $$I=\int\frac{(x^2-x+1)dx}{x^2\sqrt{x^2-x+1}}$$ and splitting we get ...
1
vote
1answer
27 views

Evaluate a difference quotient - Pre Calculus Homework

Evaluate the difference quotient: $f(x)=x^2-x+1$, $\displaystyle \frac{f(2+h)-f(2)}{h}$, $h\not=0$ I have not been able to solve this problem the farthest I have gotten to is $\displaystyle ...
3
votes
7answers
141 views

Rational Expression equivalent form

EDIT: I know how to find the answer, but does anyone know why plugging in numbers for x does not work? The Question: If the rational expression $\frac {3x^2}{3x-1}$ is rewritten in the equivalent ...
-4
votes
1answer
56 views

Two pipes cement truck time equation

Two pipes are used to pump a cement mixture out of a truck. The pipes work at the same rate. When pipe A works alone it can empty the truck in 45 minutes. When both pipes are used together they ...
0
votes
2answers
28 views

Substitute for y'

The problem was: $$ x^x=\mathrm{e}^{x-y} $$ I was able to solve it to(By Implicit differentiation) : $$ x^x\left(\ln\left(x\right)+1\right)=\mathrm{e}^{x-y}\left(1-y'\right) $$ But how do I ...
-1
votes
2answers
17 views

Co-Ordinates and midpoint [closed]

I can find the markscheme for the following questions- but they do not show the working so I am confused on how they managed to get the answers? If you would please be able to help me that would be ...
1
vote
1answer
17 views

Finding the equation for a line tangent to a parametric curve

I have the parametric equation $x = 2t - 1$ $y = 3t + 5$ $t = -1$ (defined as $t_0$) I am trying to find the line tangent to it. My book says if $x'(t_0) \not = 0$ then you can use the ...