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
votes
2answers
18 views

Algebra 2 Solving from a graph

I'm doing a practice test for my EOC which is coming up soon and I can't solve this (and I don't know what to search). How can I solve this? EDIT: Solve f(x) - g(x), to the nearest tenth, where f(x) ...
3
votes
2answers
31 views

Find $\sin\frac{\pi}{3}+\frac{1}{2}\sin\frac{2\pi}{3}+\frac{1}{3}\sin\frac{3\pi}{3}+\cdots$

Find $$\sin\frac{\pi}{3}+\frac{1}{2}\sin\frac{2\pi}{3}+\frac{1}{3}\sin\frac{3\pi}{3}+\cdots$$ The general term is $\frac{1}{r}\sin\frac{r\pi}{3}$ Let $z=e^{i\frac{\pi}{3}}$ Then, ...
0
votes
0answers
12 views

Denesting Radicals with two different radicands

After thinking some time about denesting radicals, I wondered if it was possible to denest a radical in the form $\sqrt[a]{\sqrt[b]{\alpha}+\sqrt[c]{\beta}}$ I thought about rewriting the inside to a ...
0
votes
0answers
9 views

how to find the sum of a series with finite items here?

I want to calculate $\sum_{k=i}^{N-1}\frac{1}{k-1}\frac{1}{mk+1}$, where k is not 1. Does anyone some skills?
0
votes
1answer
31 views

Why does squaring an expression with 2 subtracting terms work?

This expression can be simplified as: $$\sqrt{(x-\frac32)^2} = x - \frac32$$ Even though: $$k^2 = m^2 + n^2 \to \sqrt{k^2} = \sqrt{m^2 + n^2} \to k = \pm\sqrt{m^2 + n^2}$$ You can not remove the ...
0
votes
1answer
39 views

Rationalize the denominator: $\frac {1}{\sqrt[3]{x}+2}$

How am I supposed to rationalize the denominator for $\frac {1}{\sqrt[3]{x}+2}$? I don't even know where to really begin. I tried multiplying both the numerator and denominator by $\sqrt[3]{x}-2$, ...
0
votes
2answers
13 views

Sign of composition of transpositions

Let $\sigma \in S_n$. Definition: Suppose that $\text{sign}\sigma=(-1)^N$, where $N$ - number of inversions in permutation $\sigma$. Suppose that $\tau_1$ and $\tau_2$ transpositions. How to prove ...
1
vote
1answer
16 views

Sign of permutation. Confusing example

Let $\sigma=(2314)\in S_4$. We have different definitions of sign of permutation. 1) Our $\sigma=(24)(21)(23)$ hence $\text{sgn}\sigma=(-1)^3=-1.$ 2) Our $\sigma$ has two inversions namely $(2,1)$ ...
0
votes
1answer
12 views

GST prices, selling price and cost price

Just wondering if I have done the following question correctly. 3 people paid $459 (including GST) for a flight to Perth. Calculate the GST. I have worked that the selling price is $459 (obviously) ...
1
vote
1answer
39 views

How to solve for all real values of $x$, $y$, and $z$?

$$ \begin{align} 2z &= 1-\sqrt{x} \\ 2y &= 1-\sqrt{z} \\ 2x &= 1-\sqrt{y} \end{align} $$ Is there a way to solve it? I have tried using simultaneous equations, but it ends up with ...
11
votes
1answer
57 views

If $\log_35=a$ and $\log_54=b$, what is $\log_{60}70$?

One student sent me this question: If $\log_35=a$ and $\log_54=b$, what is $\log_{60}70$? Question asks the value of $\log_{60}70$ in terms of $a$ and $b$. Equations for $a$ and $b$ involved ...
0
votes
2answers
14 views

Cost prices, selling prices

I have a question in maths regarding GST, cost prices and selling prices. (GST is government services tax, the amount added on to an amount for the government, so it is basically tax) There are two ...
0
votes
1answer
30 views

how to get the value of x in this equation

$$n= \frac{S}{\sqrt{2\phi a}}\exp\left(-\frac{(x-R)^2}{2\phi a^2}\right)$$ I have all the values except $x$, how to get it? normally without the exponent part it easy to get $x$, I'm pretty confused ...
1
vote
1answer
28 views

Prove that the sum of the degrees in the interior angles of a polygon with $n$ sides is $180(n – 2)°$.

I would assume this question involves an inductive hypothesis. Show $n=1$ is true. Assume that if $n$ is replaced by $k$, the sum of the degrees in the interior angles of a polygon with $k$ sides ...
8
votes
2answers
93 views

Why is the Fundamental Theorem of Arithmetic so important?

I've recently read about the Fundamental Theorem of Arithmetic and I think that I have just about understood the proof. What I found quite interesting at first was the "Fundamental" part in the name. ...
0
votes
0answers
12 views

How to shift the weighted mean of a monotically increasing series of values

I have a monotonically increasing series of values $X, x\in[0,1] \ \forall i\in1,2,...,60$ the weighted mean of which is defined by $$\bar{x}=\frac{x_i+\sum_{i=2}^{60} i(x_i-x_{i-1})}{x_{60}}.$$ Given ...
-2
votes
1answer
31 views

Exponential Word Problem s(x)=5000−4000e^(−x) [on hold]

Sales of a new model of word processor are approximated by S(x)=5000−4000e^(−x), where x represents the number of years that the word processor has been on the market, and S(x) represents sales in ...
-1
votes
1answer
38 views

Exponential Word Problem $s(x)=5000-4000e^{-x}$. [duplicate]

I need help with this word problem. Much appreciated. Sales of a new model of word processor are approximated by $S(x)=5000 - 4000e^{-x}$, where $x$ represents the number of years that the word ...
-1
votes
2answers
32 views

at what interest rate can one double his initial deposit after 3 years, if the interest is compound semiannually? 24.5% 0.245% 112.25% 1.12% [on hold]

really need help with this problem. Its a little vague for me. Thanks at what interest rate can one double his initial deposit after 3 years, if the interest is compound semiannually? a.24.5% ...
2
votes
3answers
59 views

Why can't I divide this cubic through by $x$? (I see why, but what's the rule?) [duplicate]

I had this problem at school (we're just doing polynomial equations): $$2 x^3 - x^2 - 3x = 0$$ I can see that $x = 0$ is a solution to this. But I divided left and right side by $x$, factorised, and ...
2
votes
0answers
18 views

Classifying this mathematical expression

$4^{x+6}-3$ Can we say this is an algebraic expression? I believe it is an exponential expression, but is it also an algebraic expression?
0
votes
2answers
16 views

If $au + bv + cw = 0$ with $a+b + c = 0$ then $u,v,w$ are collinear

If $u,v,w \in \mathbb R^3$ such that for some $a,b,c$ real numbers with $a+b+c = 0$ we have $au + bv + cw = 0$, then why are $u,v,w$ collinear points? i substituted $a = -b-c$ and tried other things ...
3
votes
2answers
66 views

If $\sum_{n=1}^{\infty}\frac{1}{{n}^2} = \frac{{\pi}^2}{6}$ then $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$ is equal to:

If $\sum_{n=1}^{\infty}\frac{1}{{n}^2} = \frac{{\pi}^2}{6}$ then $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$ is equal to: I do not know what to try to find the solution. A hint along with the explanation ...
0
votes
2answers
20 views

How to express two variables in two other variables

If: $A=R\cos x$ and $B=R\sin x$ Then how can I express $R$ and $x$ in terms of $A$ and $B$ in a rigorous way? Meaning that I take the domain and range in account? I tried: $$\cos x=\frac{A}{R}$$ ...
0
votes
0answers
22 views

Evaluation of r^3 from zero to N (Sigma Notation)

I have to evaluate the following expression, $\sum_{n = 0}^{9} (n^3 -1)$. I know that $\sum_{n = 1}^{9}(n^3 -1)$ is given by $\frac{1}{4}(9)^2(9+1)^2 - 9$ But, how do I do this from zero to ...
1
vote
1answer
61 views

How to find $\frac{\mathrm{d}y}{\mathrm{d}x}$ when both number in front and exponent have fractions?

I'm not sure how to solve this: $\frac{5}{9}x^\frac{2}{3}$. I applied the product rule and have $\frac{2}{3}\frac{5}{9}x^{-\frac{1}{3}}$. $\frac{30}{9}x^{-\frac{1}{3}}$, then ...
0
votes
1answer
40 views

Sketching functions.

1) The functions $f$ and $g$ are as follows. $$f(x) = x^2 + 4x,\; x≥-2$$ $$g(x) = x + 6 ,\;x\in \mathbb R$$ i) Show that the equation $g\circ f(x)= 0$ has no real roots. $g\circ f(x)=(x^2 + 4x) + 6 ...
1
vote
0answers
41 views

Multiplying boths sides of an equation by $\frac{1}{x}$

I want to know what can happen if we multiply both sides of an equation by $\frac{1}{x}$, where $x$ is a variable. I mean, is it possible that we get redundant equations? Or defective equations ?
-2
votes
1answer
28 views

How to solve for a -b [on hold]

I have the relations: $$\begin{cases} a=nq_1+r\\ b=nq_2+r \end{cases}$$ How do I solve for $a - b$? Here's how I combine these equations, but I don't think it is correct: $$nq_1+r-a-b+nq_2+r=0.$$
4
votes
0answers
23 views

A function $f$ satisfies the condition $f[f(x) - e^x] = e + 1$ for all $x \in \Bbb R$.

Let $f$ be a function such that $f[f(x) - e^x] = e + 1$ for all $x \in \Bbb R$. Find $f(\ln 2)$. I've considered two cases: $f(x) = e^x + c$, where $c$ is constant. Then $f(c) = e^c + c = e + 1$, ...
1
vote
1answer
22 views

Algebraic Manipulations [duplicate]

Let a, b and c be such that $ a+b+c = 0 $ and $ l^2 = \frac{a^2}{2a^2+bc} + \frac{b^2}{2b^2+ac} + \frac{c^2}{2c^2+ba} $ The what is the value of l My approach : I could just put in the adequate ...
0
votes
1answer
29 views

If $f$ and $g$ are invertible and $f\circ g$ is defined, is $g^{-1} \circ f^{-1}$ defined?

In the proof for that any invertible functions $f$ and $g$ with $f \circ g$ defined, $(f\circ g)^{-1} = g^{-1}\circ f^{-1}$, it seems to me that there is an assumption that $g^{-1}\circ f^{-1}$ is ...
3
votes
4answers
67 views

$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$ - I keep getting imaginary numbers

$$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$$ My attempt $\sqrt{x+938^2} + \sqrt{x + 140^2} = 1116$ $(\sqrt{x+938^2} + \sqrt{x + 140^2})^2 = (1116)^2$ $x+938^2 + ...
10
votes
5answers
123 views

How do you solve $x^2 = \left(\frac 12\right)^x $?

I'm having trouble finding the steps to solve for $x$. The solutions to this equation are $x=-4$, $x=-2$, and $x=0.76666$ when solved graphically and through the solve function of a TI-nspire cx CAS. ...
0
votes
1answer
32 views

How to calculate a weighted final grade that is not out of $100\%$

I took a professor that graded his class based on weights and not points but the class weight percentages only added up to $70\%$ not $100\%$. Below is my grades: Quiz $10\%$ weight $\dfrac{374}{384} ...
3
votes
4answers
46 views

Time and Work problems

$4$ boys and $5$ girls can do $\frac{1}{2}$ work in $6$ days. after this $1$ boy and $2$ girls are added and $\frac{1}{3}$ work is done in $3$ days. how many boys must be added to complete the ...
2
votes
2answers
19 views

Proportional Distribution

I have a problem regarding supply distribution. I distribute widgets on a monthly basis; I have many customers and each of them request a different quantity each month. My monthly supply is limited ...
1
vote
1answer
31 views

Find all function that satisfy $(f(x) + xy) \cdot f(x - 3y) + (f(y) + xy) \cdot f(3x - y) = (f(x + y))^2$

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all real numbers $x$ and $y$,$$(f(x) + xy) \cdot f(x - 3y) + (f(y) + xy) \cdot f(3x - y) = (f(x + y))^2.$$
5
votes
3answers
40 views

Game is winnable if and only if $n \neq k$

Integers $n$ and $k$ are given, with $n \ge k \ge 2$. You play the following game against an evil wizard. The wizard has $2n$ cards; for each $i = 1, \ldots, n$, there are two cards labelled $i$. ...
2
votes
0answers
35 views

$(k^2)! \cdot \prod_{j = 0}^{k = 1} {{j!}\over{(j + k)!}}$ integer for $k \in \mathbb{N}$

How do I see that for any positive integer $k$,$$(k^2)! \cdot \prod_{j = 0}^{k = 1} {{j!}\over{(j + k)!}}$$is an integer?
2
votes
2answers
32 views

Domain of log function having fractional base.

Find the domain of the function $${\sqrt{\log_{0.4} (x-x^2)}}$$ Where $0.4$ is the base $0.5$ is the power on the whole bracket.
1
vote
1answer
25 views

What is the coefficient and constant term in the following sequence defined recursively?

Let $f_n(x)$ be a sequence of polynomials defined inductively as $f_1(x) = (x - 2)^2$ $f_{n+1}(x) = (f_n(x) - 2)^2$ $; n \ge 1$ Let $a_n$ and $b_n$ respectively denote the constant term and the ...
-5
votes
1answer
24 views

Finding the equation for an ellipse in $y$ form [on hold]

given $\frac{x^2}{2500} + \frac{y^2}{2500} = 1$ how do I solve for the $y=$ form of the equation? I assume there will be a $+/-$ of the same equation but have no idea how to solve
7
votes
2answers
45 views

Show that the curve $\dfrac{x^2}{a^2} +\dfrac{ y^2}{b^2} = 1$ form an ellipse

If the definition of an ellipse is the set of points $(x,y)$ such that given two focus points $F_1, F_2$ the sum of the distances from $(x,y)$ to each focus point is constant, how can one show that ...
2
votes
3answers
44 views

Precalculus/Trigonometric Functions of Sine, Cosine, and Tangent with given parameters?

for my precalculus class I was given an assignment for extra credit however it is some material that I have yet to cover or learn as far as sine, cosine, and tangent go. Below is the prompt that I was ...
0
votes
1answer
20 views

Whats the age of Bill's Kids?

I was going through some problems.This one i didn't get the solution can anyone please help me in solving this. Two old friends, Jack and Bill, meet after a long time. Jack: Hey, how are you man? ...
2
votes
2answers
78 views

We have $a^3+b^3$ and $ab$, how we can calculate $a+b$?

One of my friends is a high school student, he asked me this question. It's soluble by use of General formula for cubic roots, because: $$(a+b)^3=a^3+b^3+3ab(a+b)$$ But he looked for a simple ...
1
vote
0answers
21 views

Number of integral solutions of $x_1.x_2.x_3=x$

Let $x$ be the element of the set $A=\{1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\}$ and $x_1, x_2, x_3$ be positive integers and $d$ be number of integral solutions of $x_1.x_2.x_3=x$ , then $d$ is ...
0
votes
1answer
76 views

How to solve $2^x < x^2$

How do you solve : $$2^x < x^2$$ My math years are behind me, so I can't wrap my head around how to continue after this step : $$2^x - x^2 < 0$$ I think there's a trick since it's a 0 ...