Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

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4
votes
1answer
53 views

How to find explicit formula for a family of polynomials defined by $f_0(x)=0$ and $f_{n+1}(x)=(xf_n(x)+1)^2$?

So I have a family of polynomials defined as $$f_0(x)=0$$ $$f_{n+1}(x)=(xf_n(x)+1)^2$$ for $n\geq 0$. I am wondering if and how I would find the explicit formula for $f_n(x).$ I tried listing out the ...
1
vote
0answers
16 views

If $\frac{AM}{MB}=\frac{m}{n},$ then $AM=\frac{m}{m+n}AB$

If $\dfrac{AM}{MB}=\dfrac{m}{n},$ then why $AM=\dfrac{m}{m+n}AB?$ I am trying to show that if $M$ lies on $AB$ and $AM:MB=m:n$, then $$\vec{OM}=\dfrac{n\vec{OA}+m\vec{OB}}{m+n}.$$ For that problem I ...
-3
votes
1answer
25 views

If $a_{1}=0$, $a_{2n}=a_n+1$, and $a_{2n+1}=a_n$, then for how many $k\leq 255$ is $a_k=2$? [closed]

$a_1=0,\;a_{2n}=a_n+1\;a_{2n+1}=a_n.$ When $ a_k=2,\; (k \le 255,\ k\in \mathbb{N}).$ Find how many $k$ are there. I heard this question is related to numeral system I don't have any idea of how to ...
1
vote
4answers
49 views

Is it wrong to say that $\theta$ has to be given in “radians” (rather than “circular measure”) in, e.g., the formula $\frac12\theta r^2$?

Definitions: A radian is the measure of the central angle subtended by an arc equal in length to the radius of a circle. The SI symbol for a radian is $rad$ The circular measure of an angle is the ...
0
votes
1answer
12 views

How to divide rewards pool linearly along winners

I have a reward of 288 tokens (divisible to 8 decimal places) to distribute among the top 64 winners of a daily contest I'd like this to be fair, and use equal portions so #1 (first place) gets 64 ...
0
votes
0answers
14 views

Understanding the false positive rate of Bloom filters

I am trying to understand a scribe note on the false positive rate for a Bloom filter of size $m$ and $n$ elements. The following is stated: Suppose we are given the ratio $\frac{m}{n}$ and want to ...
2
votes
3answers
59 views

How to add (or subtract) two exponential components with same base?

I am to solve for $x$ using logs: $e^{2x}-e^x-110=0$ During my steps, I'm unsure how to combine $e^{2x}-e^x$ into one. My attempt: $$e^{2x}-e^x-110=0$$ $$e^{2x}-e^x=110$$ Here's where I get confused: $...
2
votes
3answers
69 views

A system of equations with degree 2

Let $a,b,c,d \in \mathbb R$. Suppose the following holds \begin{align} a^2-c^2 &=1 \\ b^2-d^2 &=-1 \\ ad-bc &= \pm1 \\ ab-cd &=0 \end{align} How can I find $a,b,c$ and $d$. I'm trying ...
-5
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0answers
12 views

Cardinality of set from polynomial [closed]

Let $P(x) =qx^2 +rx +s$ be a polynomial where $q, r, s \in \Bbb R$ Let $D = \{(a,b,c) : p(x) = a(x-1)^2 + b(x-2) +c \,\forall \,x \in R\}$ then cardinality of $D$ is .. plz explain simply
2
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1answer
28 views

How do I find the GCF of algebraic expressions involving negative exponents?

I'm currently reviewing college algebra and I'm learning about factoring polynomials and algebraic expressions. I have no difficulties finding the GCF of algebraic expressions whose variables have ...
0
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0answers
42 views

Proving (or disproving) an algebraic identity

It seems like a simple thing, but I am stucked on that for some time and decided ask for help...! I see on a work that \begin{align}&\dfrac{x-x^*}{x}(a-bxy-ax)+\dfrac{y-y^*}{y}(bxy-(a+c)y)\\&=...
0
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1answer
34 views

0580/42/M/J/21 IGCSE Question clarification

There was a 4 mark question on this paper that had me stumped. I don't remember it very clearly, but it was either: $3^{2y-x} = \frac{1}{9^x} \cdot 3^{2x-1}$ OR $3^{2y-x} \cdot \frac{1}{9^x} = 3^{2x-1}...
1
vote
3answers
51 views

Finding all tangent lines to circle $x^2+y^2=9$ having exactly one common point with parabola $y=x^2+3$

I have to find all tangent lines to the circle $x^2+y^2=9$, which have exactly one common point with the parabola $y=x^2+3$. I drew it and I saw that $(0,3)$ is a common point circle and parabola. I ...
2
votes
1answer
28 views

Is Blitzer Precalculus my best option considering my circumstances?

So I am wrapping up a College Algebra course from my local community college and am scheduled to start the first semester of a two semester precalculus course next month. The text book I worked out of ...
0
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1answer
33 views

If $g\circ f$ is onto then what can you say about $f$?

If $g \circ f$ is onto then $g$ will definitely be onto. But under what conditions will $f$ also be onto? Will $f$ be onto if the codomain of $g$ is the domain of $f$ ?
0
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1answer
21 views

Composite and one-one functions.

I wanted to know if $g \circ f$ is defined and is one-one then under what conditions will both $g$ and $f$ be one-one? Particularly is it possible if $f:A\to B$ and $g:B\to A$?
0
votes
2answers
63 views

Prove using binomial formula $9 \mid 10^k - 1$ for $k \in \mathbb{N}$ [duplicate]

Prove using binomial formula $9 \mid 10^k - 1$ for $k \in \mathbb{N}$ I am aware of similar answer to this question here and here however my query is about the manipulation on the binomial formula in ...
1
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0answers
44 views

Closed expression for $\sum_{k=0}^{N-1}k^2$?

I have to perform the following power sum in my embedded device $$\sum_{n=0}^{N-1}({x[n]})^2 \tag1$$ with $x[n] = 0,1,2,3...(N-1)$ Hence, I think that I can rewrite above power sum as $$\sum_{k=0}^{N-...
0
votes
2answers
34 views

Finding values where a series converges

So I was given the following prompt: "Determine the values for which the series converges" $\sum_{n=1}^\infty(-1)^n(n!)(x-3)^n$ I guess I'm a bit confused over where I'd start here, I ...
0
votes
1answer
17 views

Algebric troubles while trying to get energy of orbit in isochrone (or plummer) potential

I am facing this problem, from a 2016 Astrophysics Tripos past paper: The gravitational potential $$\Phi=-\frac{G M}{b+\sqrt{b^{2}+r^{2}}}$$ where $G$, $M$ and $b$ are constants and $r$ is the ...
0
votes
1answer
28 views

Backward Composite Function [closed]

Given that $f\circ g(x)=\dfrac{x^2-6x+2}{x+1}$ and $g(x)=1-x$, then what is $f(-1)$ equal to?
-1
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0answers
35 views

Trigonometric functions question [closed]

What's $\tan^{-1}$$(1+i)$ how can I solve it if someone help us?
0
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1answer
49 views

Factorize the following expression, $f(x) = (x-c)^3$

Take $$ f(x)=(x-c)^3 $$ How many factors does it have? I have already tried the different factorizing processeses but I do not get to the answer.
-5
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0answers
25 views

If $y = 2x^2 + bx + 1$ has a turning point at $x = 1.7$, then what is value of $b$?

I need help on this math question: If $y = 2x^2 + bx + 1$ has a turning point at $x = 1.7$, then what is value of $b$?
1
vote
1answer
55 views

Prove that $\frac{1}{kn} + \frac{1}{kn + 1} + \dotsb + \frac{1}{kn + n - 1} > n \left(\sqrt[n]{\frac{k+1}{k}} - 1 \right)$

Let $k,n \in \mathbb{Z}^+$ with $n > 1$. Prove that $$\frac{1}{kn} + \frac{1}{kn + 1} + \dotsb + \frac{1}{kn + n - 1} > n \left(\sqrt[n]{\frac{k+1}{k}} - 1 \right)$$ I roughly observe that AM-...
-7
votes
0answers
20 views

Random question with unknowns?

$\frac{n1}{p1} = a$, $\frac{n2}{p2} = b$, $\frac{n3}{p3} = c$, $\frac{n4}{p4} = d$. $0.5(a + b) = k1$, $0.5(c + d) = k2$ $n1 + n2 = N1$, $n3 + n4 = N2$, $p1 + p2 = P1$, $p3 + p4 = P2$ If $\frac{N2}{N1}...
3
votes
4answers
58 views

For positive integer $n$, why is $\lfloor \log_{10}(2^n)\rfloor + \lfloor \log_{10}(5^n)\rfloor + 2 = n+1$?

For positive integer $n$, why is $\lfloor \log_{10}(2^n)\rfloor + \lfloor \log_{10}(5^n)\rfloor + 2 = n+1$? This question comes from counting the number of digits of $10^n$ in terms of the number of ...
2
votes
1answer
45 views

What is the difference between the graphs of $z=xy$ and $z=x^2-y^2$?

What is the difference between the shape of the graphs of $z=xy$ and $z=x^2-y^2$? Are they different or just rotated? Kind of hard to confirm, because I know you can transform them. But I don't know ...
1
vote
1answer
28 views

Are the following systems of inequalities the same?

Suppose $x, y \in \mathbb{R}$, and $\mathcal{S_1}$ is a system of inequalities: \begin{align*} \mathcal{S_1} &= \begin{Bmatrix} x - y \geq 1\\ -x + 2y \geq 1\\ 3x - 5y \geq 2 \end{Bmatrix}\\ &...
-2
votes
0answers
40 views

Functions, please help with question number 4. [closed]

A real estate company owns the Jardines del Maule apartment complex, which comprises 96 apartments. If the rent is $ 300,000 per month, all the apartments are occupied. However, for every $ 30,000 per ...
2
votes
0answers
45 views

What techniques exist for working with “iterated” functions such as f(f(x))?

I keep encountering questions that sound like obscure puzzles, involving expressions such as $f(f(x))$. I don't know of any techniques for working with such things, or getting from knowledge of $f(f(...
1
vote
1answer
11 views

Implicit differentiation rearrangement

I had to calculate $\frac{dy}{dx}$ for the equation $2x^2 = \frac{x+y}{x-y}$ If I rearrange the equation like this: $2x^2(x-y) = x+y$ Now when I calculate $\frac{dy}{dx}$ for the above 2 equations, I ...
0
votes
1answer
58 views

Express given cos and sin functions in the complex form $\Re\left(re^{i\theta}\right)$

$ f(x) = \sqrt{2}\sin\left(\frac{2\pi}{3}+5x\right) $ and $g(x) = \sqrt{3}\cos\left(\frac{\pi}{4}+5x\right)$. Express the functions $f(x)$ and $g(x)$ in the complex form $\Re\left(re^{i\theta}\right)$....
0
votes
3answers
39 views

If $x^2=10z-34$, $y^2=8x-23$, $z^2=7-6y$, find the integer value of $x+y^2+z^3$

The following set of equations is given: $$\begin{align} \ & x^2=10z-34 \\ \ & y^2=8x-23 \\ \ & z^2=7-6y \end{align}\\$$ It is asked to find the integer value of $x+y^2+z^3$. I have ...
2
votes
3answers
53 views

Rewrite $\frac{125}{\left(\frac{1}{625}\right)^{-x-3}}=5^3$ in a common base then solve for $x$

I am to rewrite $\frac{125}{(\frac{1}{625})^{-x-3}}=5^3$ and then solve for x. My textbooks solutions section says the solution is -3. I gave it a shot and got 3.25. Here is my working: $$\frac{125}{\...
1
vote
1answer
37 views

Spivak Chapter 4 Question 7

I'm a bit confused about this problem, from chapter 4 of Spivak's Calculus. In particular, I'm not sure what a straight line is defined as in Spivak. Earlier in the text, Spivak defines a straight ...
1
vote
0answers
29 views
+50

Replication invariance of weighted sums

Consider the following function: $\phi(\langle x_1,x_2,...,x_n \rangle) = \alpha_1 x_1 + \alpha_2 x_2 + ... + \alpha_n x_n$, where $x_1, x_2, ...$ are all real numbers such that $x_1≤x_2≤...$ and ...
1
vote
1answer
64 views

Prove that $\frac{1}{1⋅2}+\frac{1}{3⋅4}+\frac{1}{5⋅6}+…+\frac{1}{199⋅200}$ = $\frac{1}{101}+\frac{1}{102}+\frac{1}{103}…+\frac{1}{200}$ [duplicate]

Prove that $$\frac{1}{1⋅2}+\frac{1}{3⋅4}+\frac{1}{5⋅6}+....+\frac{1}{199⋅200}= \frac{1}{101}+\frac{1}{102}+\frac{1}{103}...+\frac{1}{200}$$ My Approach: $T_{r}=\frac{1}{\left(2r\right)⋅\left(2r-1\...
0
votes
2answers
81 views

Confusion in BODMAS rule to evaluate $\left(\frac15\right)\div\left(\frac15\right)\div\left(\frac15\right)\div\left(\frac15\right)$ [duplicate]

I am stuck with a question $$\left(\frac15\right)\div\left(\frac15\right)\div\left(\frac15\right)\div\left(\frac15\right)$$ We can have different approach to this problem but how are we gonna apply ...
-1
votes
2answers
41 views

Set of all points $(x,x-1)$ where $x \in \mathbb{Q}$ is countable

I am able to see that the set of all points $(x,x-1)$ where $x \in \mathbb{Q}$ is countable since it is a subset of $\mathbb{Q} \times \mathbb{Q}$. But I am finding difficulty in finding a one-one ...
0
votes
0answers
36 views

Linear regression coefficient changes?

If I have a linear regression model described by $y = 0.47x$, then I know that a one unit increase in $x$ translates to a $0.47$ unit increase in $y$. Can I also say that a $0.1$ unit increase in $x$ ...
0
votes
1answer
33 views

Roots and squares in log functions

This is the question I'm trying to solve right now. I know since $x$ is on numerator and $y, w$ are in denominator, they subtracted $y$ and $w$ from $x$. What got me confused is that I have a dim ...
-5
votes
0answers
24 views

What is the maximum revenue if I am given the cost fuction and the revenue function? [closed]

If the total cost for a product is given by C(x) = 900 + 25x and the total revenue is given by R(x) = -x2 +100x
-3
votes
1answer
37 views

What is the maximum profit if I am given the cost fuction and the revenue function?

Maximum Revenue $$ C(x)=1600+1500x $$ $$ R(x)=1600x−x^2 $$
0
votes
3answers
70 views

Proving $\binom{2n}{n}>\frac{2^n}{\sqrt{n\pi}}$ [closed]

I want to prove the following inequality $\binom{2n}{n}>\frac{2^n}{\sqrt{n\pi}}$ by induction. Can anyone suggest a hint?
-4
votes
1answer
33 views

Properties of max(x,y) operators [closed]

Does $\max (x_{1}+c,x_{2}+c,x_{3}+c,...,x_{n}+c)$ = $=\max (x_{1},x_{2},x_{3},...,x_{n}) +c$ where $x > 0$ and $c > 0.$
0
votes
2answers
46 views

Solving $x = 5(y^2+10y+20)$ for $y$

I have an equation: $$x = 5(y^2+10y+20)$$ For context, $y$ is a value that represents the current level of a user. The equation is looking for $x$, i.e finding how much experience ($x$) is required to ...
2
votes
3answers
126 views

Minimizing $a_1x_1^2 + a_2x_2^2$ for positive $a_i$, where $a_1x_1+a_2x_2=B$

Find $$\min\{a_1x_1^2 + a_2x_2^2\}$$ Where $ a_1x_1 + a_2x_2 = B$, and $a_1>0$ and $a_2>0 $. Find $x_1$ and $x_2$. Can we do it usig AG mean inequality? Let's say we have the problem to find ...
0
votes
4answers
51 views

Minimizing the sum of the distance of two points and a point on an ellipse

Consider coordinates $B=(-16,0)$ and $A=(12,8).$ Let $D=(x,y)$ be a coordinate of the ellipse $9x^2+25y^2=3600$ such that the distance of $AD+DB$ is minimized. For what value of $(x,y)$ is $AD+DB$ ...
0
votes
1answer
41 views

While changing limits on integral, when using substitution , both upper and lower limits comes same, does this mean the value of integral is zero?

When i did that substitution both upper and lower bound comes out to be same, i have solved the integral using trigonometry and ans is not zero.

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