# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f$ ?
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### 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 $g:B\to A$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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}\\ &...
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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 ...