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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0answers
17 views

Algebra Integral simplification

Let some equation problem final result is like this \begin{align} M=\mathrm{exp}\bigg\{-\pi\lambda v^2+\pi\lambda v^2\bigg(\displaystyle\int_o^s \frac{2x}{v^2}\mathrm{d}x \nonumber \\\\ ...
2
votes
3answers
28 views

A not so hard basic calculus problem? But it appears to be very lengthy

Find the coordinates of the two points on the curve $y=4-x^2$ whose tangents pass through the point $(-1,7)$. My work: Let the two points be $(a,b)$ and $(c,d)$. And $\frac{dy}{dx}=-2x$, so the ...
3
votes
2answers
68 views

How can I prove that $2ab \leq a^2 + b^2$?

I'm stuck with it: $2ab \leq a^2 + b^2$. Have no idea how to go beyond this ($a,b \geq 0$). Thanks!
-2
votes
2answers
17 views

Pricing call options with binomial trees (proof) [on hold]

I need assistance in proving that the following line: $$f = S_0\left(\frac{f_u - f_d}{S_0u - S_0d}\right)\left(1 - ue^{-rT}\right) + f_ue^{-rT}$$ Equals this line: $$f = \frac{f_u\left(1 - ...
0
votes
4answers
29 views

Question on polar coordinates and cartesian coordinates

I know the conversion between polar coordinates and cartesian coordinates. Nevertheless, I cannot understand why $r=2a\cos\theta$ represents a circle of radius $a$ and center $(a,0)$. Can anyone ...
1
vote
2answers
25 views

Function Composition Thinking Problem

Here is the question: A banquet hall charges $\$975$ to rent a room, plus $\$39.95$ per person. Next month they will offer a $20\%$ discount off the total bill. Determine two equations, one for ...
0
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2answers
28 views

Finding two functions $f(x)$ and $g(x)$

I am not sure how to approach this question. It asks to find $f(x)$ and $g(x)$ such that $h(x)=f(g(x))$, for each function: a) $$h(x)=\sqrt{x^2 + 6}$$ b)$$h(x)=\frac{1}{x^3}-7x+2$$ If someone ...
0
votes
1answer
21 views

Determine the value of combined functions with square roots

The question I have is to determine the value of $f(g(x))$ given $f(x)=\sqrt{16-x^2}$ and $g(x)=x^2$ I know generally how to tackle these kinds of questions, but I am not sure what to do when there ...
0
votes
3answers
23 views

Find the domain of combined functions

I have a question asking to find the domain of $g(f(x))$ given $f(x)=2x^2+x$, and $g(x)=x^2+1$. I can easily do these questions in reverse when you have to find $f(g(x))$, but when having to find ...
1
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0answers
32 views

Solve $x=C \log(C \log(x+A)+B)$

Is it possible to resolve an equation of the type $$x=C\log{(C\log{(x+A)}+B)}$$ (where $A$, $B$, and $C$ are real-valued parameters) for $x$? As far as I can see, the function on the right hand ...
0
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0answers
20 views

free tool for algebraic manipulations of commutator expressions

Is there an (ideally) free tool for algebraic manipulations of commutator expressions of the form: given $$c(A,B):=\tfrac{1}{2}[A,B]+\tfrac{1}{12}[A,[A,B]]+\tfrac{1}{12}[B,[B,A]]$$ simplify (express ...
3
votes
1answer
79 views

Guessing the other root to a quadratic equation

I just attempted to do the question below, but it seems that even after seeing the answer I'm not sure I understand the motivation for the solution. Let $\alpha ...
0
votes
1answer
52 views

Calculate the sum $\sum^{\infty}_{j=1}(2\sqrt{2}-3)^j$

Would appreciate if anyone could help with the summation \begin{equation*} \sum^{\infty}_{j=1}(2\sqrt{2}-3)^j. \end{equation*} Thanks a lot.
1
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0answers
25 views

Polynomial With Complex Zeros

There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let ...
1
vote
1answer
9 views

Sequence of functions that extends the algebraic properties of exponents to higher level operators.

I was thinking about some simple algebraic exponent properties such as the following $$ z^{x+y} = z^xz^y $$ and I started wondering about analytically continuing this identity to "higher-level ...
0
votes
4answers
52 views

Problem Verifying Two Challenging Trig Identities

My math teacher gave us an equality involving trigonometric functions and told us to "verify" them. I tried making the two sides equal something simple such as "1 = 1" but kept getting stuck. I would ...
1
vote
3answers
48 views

Use quadratic formula to find upper and lower limits of an expression

Using quadratic formula show that $\frac{x^2-x+1}{x^2+x+1}$ lies between $3$ and $\frac{1}{3}$ for all real values of $x$. Let $\frac{x^2-x+1}{x^2+x+1}=y$, then ...
1
vote
2answers
36 views

Solving $x^y = y^x$ analytically in terms of the Lambert $W$ function

I'm interested in deriving the solution for $y$ in terms of $x$ given $x^y = y^x$ using the Lambert $W$ function. Wolfram Alpha states: $$y = - \frac{x\ W\left(-\frac{\log(x)}{x}\right)}{\log(x)}$$ ...
0
votes
2answers
28 views

If $\sin s=-1/3$ and $s$ is in the $4$-th quadrant, find the exact value of $\sin (2s)$ [on hold]

Could someone solve this step by step so I can wrap my head around the process?? If $\sin s=-1/3$ and $s$ is in the $4$-th quadrant, find the exact value of $\sin (2s)$.
0
votes
1answer
15 views

Partition set of $n$ elements until each partition contains $1$ element. Must terminate after exactly $n-1$ iterations?

Suppose I have a set of $n$ elements and I want to partition the set (split into two) until each partition contains a single element. How do I see that the terminating case must occur after exactly ...
2
votes
1answer
38 views

Prove that $1/(\sin x + 1) - 1/(\sin x - 1) = 2 \sec^2 (x)$

Can anyone solve this for me? Prove that $\frac1{\sin x + 1} - \frac1{\sin x - 1} = 2 \sec^2 (x)$. This is as far as I went: $$\frac{(sin x - 1) - (sin x + 1)}{(sin x + 1)(sin x - 1)}$$ ...
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1answer
21 views
-1
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2answers
46 views

Show that if x>0, x+1/x >= 2. [on hold]

How am I to prove this inequality without use of calculus: for any real x>0, x+1/x >= 2 ? Thanks for any help.
0
votes
3answers
55 views

Prove that $\alpha$ lies between $0$ and $4$.

Let $a,b,c$ be the length of the sides of the triangle $ABC$ . Given $(a+b+c)(b+c-a)=\alpha bc$.Then Prove that the value of $\alpha$ lies in between $0$ and $4$. ...
0
votes
5answers
69 views

simplify the following rational expression

Simplify the following $$ \frac{x^2-x-2}{x^2-3x} \times \frac{x^2-x-6}{x^2+5x+4} $$ I don't know how to approach it. I tried doing the quadratics first but now I'm stuck after getting $$ ...
2
votes
3answers
50 views

Do asymptotes disprove 0.9 repeating equal 1?

I am in 9th grade and taking geometry. Several of my friends taking pre-calc say that 0.9999... does not equal 1, but is just an asymptote. I have not taken that subject yet and they don't give any ...
-1
votes
4answers
24 views
-3
votes
2answers
34 views

Find minutes when digit sum is 20? [on hold]

When a digital clock reads 3:47, the sum of digits is 14. How many minutes after 3:47 sum of digits will be 20 for first time? a) 42 b) 132 c)192 d)251 ...
0
votes
2answers
40 views

Trigonometric double angle formulas problem

I want to simplify the answer to an equation I had to compute, namely, simplifying $\sin^2 (2y) + \cos^2 (2y)$. I know that $\sin^2 (y) + \cos^2 (y) = 1$ but is there anything like that I can use at ...
2
votes
1answer
39 views

Solve $z + z^{-1} = x$ with elementary methods

I want to solve the equation$$z + z^{-1} = x\tag{1}$$ with elementary methods. I know the two solutions to be $$x_{1}=\frac{1}{2}x+\frac{1}{2}\sqrt{x^{2}-4}\qquad\text{or}\qquad ...
5
votes
4answers
127 views

How to solve an equation with $x^4$?

Today, I had this question on a Maths test about Algebra. This was the equation I had to solve: $$(1-x)(x-5)^3=x-1$$ I worked away the brackets and subtracted $x-1$ from both sides and was left with ...
1
vote
1answer
36 views

Finding an nth degree polynomial.

First post here. So I'm having a bit of trouble with the eponymous question type. It's a bit embarrassing, as the problem is almost purely conceptual in nature, and I thought I had basically ...
2
votes
6answers
77 views

Solving $e^\frac1x = x$ non-graphically?

This question has come up twice in different tests and the instructions always point out that it should be solved using a graphic calculator. Fair enough, the answer is ≈ 1.76322...(goes on forever?). ...
1
vote
1answer
16 views

Compound Interest Calculation

In __________ years a sum will double at $5\%$ per annum compound interest. Options given are: a. 15 years 3 months b. 14 years 2 months c. 14 years 3 months d. 15 years 2 months The way to ...
6
votes
3answers
826 views

A supposed to be easy calculus problem

Find the values of $m$ if the line $y=mx+2$ is a tangent to the curve $x^2-2y^2=1$. My working: First we differentiate $x^2-2y^2=1$ with respect to $y$ to get the gradient. We get ...
0
votes
2answers
26 views

Find a function $g(x)$ satisfying the above conditions.

Find a function $g(x)$ satisfying the above conditions:- a)domain is $(-∞,∞)$. b)range is $[-2,8]$. c)$g(x)$ has a period $π$. d)$g(2)$=3. ATTEMPT: Since the function is periodic with period $π$ ...
1
vote
4answers
41 views

A line through the point P(8, -7) is a tangent to the circle C at the point T. Find T

Circle C equation $(x+5)^2+(y-9)^2=25$ A line through the point P(8, -7) is a tangent to the circle C at the point T. Find T. I tried simultaneous equations: 1. $(x+5)^2+(y-9)^2=25$ 2. $y = ...
0
votes
0answers
37 views

Law of Cosines for SSA triangles

In most geometry courses, we learn that there's no such thing as "SSA Congruence". That is, if we have triangles $ABC$ and $DEF$ such that $AB = DE$, $BC = EF$, and $\angle A = \angle D$, then we ...
0
votes
1answer
20 views

Does the function have horizontal or vertical asymptotes?

So I'm analyzing some functions here and I need to determine whether or not they have horizontal or vertical asymptotes. The equations are: $f(x)=260$ $g(x)=1+24(0.9)^x$ $h(x)=f(x)/g(x)$ Now ...
10
votes
1answer
184 views

Solving a special Quartic Equation.

Solve for $x$ $$(x^2-4)(x^2-2x)=2$$ I have tried the Rational Root Theorem and found that there are no rational roots. Further, the polynomial $p(x)=(x^2-4)(x^2-2x)-2$ is irreducible since ...
-2
votes
2answers
37 views

Having difficulty understanding polynomial equation [on hold]

I'm not sure how to factor this: $27x^3 - 64$. It deals with polynomial. I've never worked polynomial expressions before, so If anyone could help me figure out how to work with polynomials, that would ...
0
votes
3answers
41 views

The inequality $k(n-1)<n^2-2n$ for all odd $n$ and $k<n$

How one can prove the following statement: $k(n-1)<n^2-2n$ for all odd $n$ and $k<n$ Tried so far: induction on $n$, graphing, and rewriting $n^2−2n$ as $(n−1)^2−1$.
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2answers
12 views

Horizontal and Vertical Asymptotes of functions

So I'm completing a chart analyzing the different properties of three different functions: $f(x)=\log(x^2+6x+9), g(x)=\sqrt{x^2 -1}$ (sorry not sure how to do square roots on here), $h(x)=f(x)(g(x))$ ...
1
vote
2answers
26 views

If you fold a rectangular piece of paper in half [duplicate]

If you fold a rectangular piece of paper in half and the resulting rectangles have the same aspect ratio as the original rectangle, then what is the aspect ratio of the rectangles?
7
votes
2answers
134 views

Polynomial Functional Equation.

Let $f(x)$ be a one-one, polynomial function such that $f(x)f(y)+2=f(x)+f(y)+f(xy) \ \forall \ x,y \in \mathbb R - \{0\}$, $f(1) \neq 1$, $f'(1)=3$. Find $f(x)$. I tried to find the degree of ...
4
votes
3answers
311 views

“Rationalizing” an equation

$$x=\sqrt[3]{p}+\sqrt[3]{q}$$ I'm trying to figure out some way to "rationalize" the previous equation, meaning to rewrite it purely in terms of whole number powers of $p$, $q$, and $x$. It seems ...
1
vote
4answers
40 views

A line through the point P(8, -7) is a tangent to the circle C at the point T. Find the length of PT.

Circle C equation $(x+5)^2+(y-9)^2=25$ A line through the point P(8, -7) is a tangent to the circle C at the point T. Find the length of PT. The question itself is easy when using pythagoras, ...
0
votes
0answers
34 views

How to find the minimum value of an epression if a constraint is given? [duplicate]

Given that $x,y,z$ are positive real numbers satisfying the equation $$xyz=32$$ find the minimum value of $$x^2+4xy+4y^2+2z^2.$$
3
votes
2answers
134 views

Basic Logarithm question - I can't get both answers from quadratic

Here's the Question : If $xy$ = $64$ and $\log_x y + \log_y x = \frac{5}{2}$, find $x$ and $y$ I can get this to $$log_x y + \frac{1}{\log_x y} \frac{5}{2}$$ let $\log_x y = N$ $$N + ...
0
votes
4answers
46 views

How to find : Range of $cos(cosx) $

How to find : Range of $\cos(\cos x) $ My approach : Since $-1 \leq \cos x \leq 1$ we get $\cos(-1) \leq \cos(\cos x) \leq \cos(1)$ Is it correct? please suggest. Can we use the above method ...