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
3answers
27 views

Why is the limit of this graph not 4.3?

I just took an online exam as part of a Precalculus course, and one of the problems on my test was as follows: Estimate $\lim \limits_{x \to 2} f(x)$ from the graph below. The available ...
4
votes
4answers
48 views

How do you factor $\frac{2x^2-x-1}{x^2-9} \cdot \frac{x+3}{2x+1}=$?

\begin{align} & \frac{2x^2-x-1}{x^2-9} \cdot \frac{x+3}{2x+1}= \frac{2x^2-x-1}{(x-3)(x+3)} \cdot \frac{x+3}{2x+1} \\[10pt] = {} & \frac{2x^2-x-1}{(x-3)} \cdot \frac{1}{2x+1}= ...
-5
votes
3answers
29 views

one number is six more than another.The sum of the two numbers is 28. What are the numbers? [on hold]

one number is two more than another.The sum of the two numbers is 20. What are the numbers?
0
votes
2answers
20 views

Need help to understand a math task about algebraic and parametric equations

Can anybody please explain this for me?: Find the algebraic and parametric equations of the circle with centre (-2,3) that passes through (1,-1) How do I find the algebraic and parametric ...
2
votes
3answers
200 views

How to find out which number is larger without a calculator?

So I have a question which is: Which is larger? $$2.2^{3.3} \text{ or } 3.3^{2.2} $$ Now I need to find out with using a calculator but the answer is $3.3^{2.2}$. The only thing I could think of ...
-4
votes
1answer
22 views

Word problem including time [on hold]

Two vans are 420 miles apart when they begin traveling toward each other. One can goes 50 mph while the other goes 55 mph. How long do they drive before they meet?
-1
votes
3answers
26 views

Guaranteed Profit on a Roulette wheel: Does $N$ exist?

I'm looking for a good answer that can be explained to someone without much maths. If a casino spins a roulette wheel enough times, the probability the casino makes a profit $\to$ 1 as $N \to \infty$ ...
5
votes
4answers
105 views

How many ways to write $2010$?

Let $ N$ be the number of ways to write $ 2010$ in the form $ 2010 = a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0$, where the $ a_i$'s are integers, and $ 0 \le a_i \le 99$. An example of ...
0
votes
2answers
33 views

Inverse functions: what is the difference between $\tan^{-1}(x)$ and $\tan(x)^{-1}$?

I’ve never really been taught about inverse functions, and I figured this is a pretty simple question, but I couldn’t find any explanation in my math textbook about this. What is the difference ...
0
votes
3answers
28 views

Trying to show $|\overrightarrow{a}\times\overrightarrow{b}|^2=|\overrightarrow{a}|^2|\overrightarrow{b}|^2-(\overrightarrow{a}⋅\overrightarrow{b})^2$

If $\overrightarrow{a} = \langle a_1, a_2, a_3 \rangle$ and $\overrightarrow{b} = \langle b_1, b_2, b_3 \rangle$, then the cross product of $\overrightarrow{a}$ and $\overrightarrow{b}$ is the ...
2
votes
0answers
14 views

Generalised equation/Notation for writing down products of sets of combinations

I am trying to write a generalized equation to solve a fairly simple probability problem (c & k are constants) $$y_{1} = (1 - cx_{1})^k$$ $$y_{2} = \frac{(1 - cx_{2})^k - ...
2
votes
1answer
42 views

Simplified is 0? $\log_{3} 9x^4 - log_{3}(3x)^2 $

I have tried to solve this multiple ways, but I keep getting $2\log_{3}x$. According to the answer key, it is supposed to work out to 0, but I'm not seeing it. Can someone point me in the right ...
1
vote
1answer
38 views

Convert from base 10 to base 5

I understand the integer part, keep dividing by 5 and I get 112, but for the fraction part I need a help. The number is: $$ (32.\bar 5)_{10} = (112, ??)_{5} $$ $$ 0,5 * 5 = 2,5\\ 0,5 * 5 = 2,5\\ ...
-7
votes
0answers
35 views

Prove the following statement by contradiction: [on hold]

Prove the following statement by contradiction: if $x,y >0$ then $(x + y)^2 \ne x^2 + y^2$.
0
votes
0answers
23 views

Range edge attraction of strength n [on hold]

Basically I want a function that takes a domain $m$ that goes from [0, $n$] where $n$ is a real number, a position/value $x$ within that domain, and an edge attraction strength $t$. If I say ...
2
votes
0answers
33 views

Algebric equation problems

Find $a, b, c \in \Bbb Q$ such that: $\sqrt[3] a + \sqrt[3] b + \sqrt[3] c = \sqrt[3]{\sqrt[3] 2 - 1}$.
0
votes
1answer
32 views

Problem of Mathematical Induction [on hold]

Show that $n!\leq 2^{-n}(n+1)^n$ for all $n\in\mathbb{N}$ and equality holds if and only if $n=1$.
-3
votes
2answers
74 views

Solve equation $x^3 - 3x = \sqrt{x + 2}$ [on hold]

Solve :  $$x^3-3x=\sqrt{x+2}$$ For any real number x.
0
votes
0answers
17 views

Additive exponential function

I am trying to solve the following for $x$: $$ ax = b(P-P^x)$$ where $a$, $b$ are positive constants, and $P \in (0, 1)$. I already know that the left hand side is linear and the right hand side ...
-3
votes
1answer
15 views

ATV business and percentages [on hold]

I have 12 ATVs. My friend owns 2 of them. Through rental, we make $2,020$ \$. What percent of that amount is of my friend? How much is it?
0
votes
1answer
10 views

Modulus of Z (Normal distribution)

The random variable $Z$ is distributed such that $Z \sim N(0,1)$ find the probability of $P(\left|Z\right| >2.4)$. How to solve this modulus type of question ?
2
votes
2answers
21 views

Finding $P$ knowing $\overrightarrow{PQ}×\overrightarrow{b}$, $\overrightarrow{PQ}⋅\overrightarrow{c}$, $\overrightarrow{b}$, and $\overrightarrow{c}$

Let $Q$ be the point $(1,2,3)$, let $\overrightarrow{b} = \langle -1, 0, 1\rangle$, and let $\overrightarrow{c} = \langle 2, 1, 5\rangle$. It is known that $\overrightarrow{PQ} \times ...
1
vote
2answers
40 views

Equality in AM GM Inequality

In AM GM inequality for nonnegative real numbers $a_1,a_2,\ldots,a_n$, How to show that if equality holds then $a_1=a_2=\ldots=a_n,$ using method of induction?
0
votes
0answers
9 views

Why is Distribution Prioritized Over Combining?

In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [a(b + ...
0
votes
0answers
15 views

What are the loci of points z which satisfy the following relations?

a) |Z-Z1|=|Z-Z2| b) 0< Re(iZ)<1 c) |z|=ReZ+1 d) Im((Z-Z1)/(Z-Z2))=0 The professor did not explain loci in class and the text does not have any examples, so I am completely lost.
0
votes
2answers
19 views

How do I complete composite functions that use cos(x) and a polynomial? [on hold]

Let $f(x)= \cos(x)$ and $g(x)= 3x^3+4x^2-7$. Find $g(f(x))$ and $f(g(x))$
-1
votes
0answers
16 views

simplifying distance equation

The optimal angle for throwing a ball from a cliff is $$\sin \theta = \frac{1}{(2+ \beta)^{1/2}}$$ the original distance equation is $$ x = \cos \theta (\sin \theta + (\sin^{2} \theta + ...
2
votes
4answers
39 views

Solving for b for equation $3a−5=−4b+1$ [on hold]

I have a math problem, and I'm trying to solve for 'b'. The problem answer shows that the first step is going from Step 1 to Step 2. I don't understand how they are doing this. How does the $-4b$ ...
4
votes
2answers
60 views

Why $\tan x>\sin x$ in this question?

The question asks me to prove the identity $\tan ^2x-\sin ^2x=\tan^2 x \sin^2 x$ and use this result to explain why $\tan x>\sin x$ for $0<x<90$ I've proved the identity and I can't explain ...
2
votes
2answers
45 views

How do you factor $3x^{3/2} -9x^{1/2}+6x^{-1/2}$?

How do you factor $3x^{3/2} -9x^{1/2}+6x^{-1/2}$ ? I factored out a 3 to get: $3(x^{3/2} -3x^{1/2}+2x^{-1/2})$, but it seems this can be factored further.
0
votes
2answers
31 views

Domain and range of $f(x,y)=\sqrt{1+x-y^2}$

I need to find the domain and range of $f(x,y)=\sqrt{1+x-y^2}$. Can someone walk me through the proper reasonings in solving this problem? My attempt Domain From looking at the function I get: ...
2
votes
1answer
31 views

How do you factor $x^3-3x^2-4x+12$

How do you factor $x^3-3x^2-4x+12$ ? I tried to factor $x(x^2-3x-4) + 12$ instead and I got $x(x-4)(x+1)+12$ but apparently this can be factored further.
3
votes
1answer
31 views

Largest Triangular Number less than a Given Natural Number

I want to determine the closest Triangular number a particular natural number is. For example, the first 10 triangular numbers are $1,3,6,10,15,21,28,36,45,55$ and thus, the number $57$ can be ...
-5
votes
2answers
24 views

Modifying a recipe, changing it from 8 to 10 servings.

A recipe that makes 8 servings calls for 3/5 cup flour. Jeff modifies the recipe so that it can serve 10. How many cups of flour does he need?
3
votes
4answers
87 views

Why doesn't quadratic formula lead to a the correct factored form of the original equation?

Applying the quadratic formula to $2x^2-3x+1$ we have \begin{eqnarray*} a&=&2 \\ b&=&-3 \\ c&=&1 \end{eqnarray*} which gives me two roots: \begin{eqnarray*} x_1&=&1 ...
2
votes
4answers
65 views

Why does the a*c cheat work when factoring trinomials?

When factoring a trinomial, in the form $ax^2 + bx + c$, I am told that one can multiply $a$ and $c$ which gives a product whose factors add to $b$. So if I have $2x^2 + 5x -3$ that gives me $-6$. ...
-2
votes
1answer
33 views

Solve the quadratic equation $(2-y)^4=3(2-y)^2+1$

Solve $$(2-y)^4=3(2-y)^2+1$$ The answer is supposed to be $y=4\pm \sqrt{6+\frac{13}2}$. I have tried to work this problem out but I cannot get the answer that is in the book.
-1
votes
1answer
24 views

Adding two variables with subscripts [on hold]

What is the explanation to why $x_{3k} + x_{3k+1}$, is equal to $x_{3k+2}$. Isn't that incorrect because there is no value 1 in the subscript $x_{3k}$? I saw this in a prove in ...
0
votes
1answer
26 views

Two variables in one equation

I am currently having some trouble getting through the following exercise: "There are $25$ apples in a basket in which teacher eats an $X$ amount of them and gives the remaining apples to Y amount of ...
22
votes
1answer
451 views

Parabolas in sequences of digits from the Fibonacci sequence

In preperation for an exam, I was studying Haskell. Therefore I was solving an old assignment where you had to define the fibonacci series. After solving the task (see 1] for source code) and ...
4
votes
4answers
78 views

Calculate simple expression: $\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}$

Tell me please, how calculate this expression: $$ \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} $$ The result should be a number. I try this: $$ \frac{\left(\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - ...
0
votes
0answers
26 views

For the classical diffusion equation ut = r (5ru) (in 3 space dimensions)

fi nd TWO changes of variables which changes the di ffusion constant from 5 to D = 1 for the new coordinate system?
8
votes
4answers
668 views

Simple Trig Equations - Why is it Wrong to Cancel Trig Terms?

In the following problem, I first did it using a cancellation of $sin^2\theta$, working shown below, which gave the wrong answer. Having looked at the question again, I saw it could be solved by ...
-2
votes
1answer
29 views

How do you find an expression for the sum of the first 35 terms of a logarithmic series? [on hold]

$$\ln(x^2/y^0) + \ln(x^2/y^1) + \ln(x^2/y^2)+ \ln(x^2/y^3)+ \ln(x^2/y^4)+ \cdots$$
0
votes
0answers
21 views

Parametric vector form of cartesian equation

Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1 ...
-1
votes
0answers
29 views

Please solve this algebra question [on hold]

2(24-____)=____-34 I really don't know who to do this! I really need some help D;!
0
votes
3answers
47 views

prove $x\le y$ when $x<z$ for every $z>y$ [on hold]

Let $x,y$ be real numbers. Prove that if $x\le z$ for every $z>y$, then $x\le y$.
2
votes
2answers
52 views

If $a,b,c,d,e,f$ are non negative real numbers such that $a+b+c+d+e+f=1$, then find maximum value of $ab+bc+cd+de+ef$

$(a+b+c+d+e+f)^2=$ sum of square of each number (X)+ $2($ sum of product of two numbers (Y) $)$ $ab+bc+cd+de+ef \le Y$ since all are positive. Therefore $1\ge X+(ab+bc+cd+de+ef)$ Edit: From AM GM ...
2
votes
2answers
40 views

How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$?

How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$? In my title, I should be clear that the additive inverse should be unique. But how does this help? I dont even get why ...
-2
votes
4answers
41 views

Find two numbers knowing their sum and their difference [on hold]

The sum of two numbers is 15 and their difference is 3. What are the numbers and their product?