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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.

This tag is for questions typically taught in precalculus, as well as elementary algebra.

These topics include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomials, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

47234 questions
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Proving that $u_n $ is arithmetic sequence

Let $u_n$ be a sequence defined on natural numbers (the first term is $u_0$) and the terms are natural numbers ($u_n\in \mathbb{N}$ ) We defined the following sequences: $$\displaystyle \large x_n=u_{u_n}$$ $$\displaystyle \large y_n=u_{u_n}+1$$ If…
Brab
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Does $X^{2/2} = X^{1/1}$?

I'm having a bit of a hard time wrapping my head around how the following that I have just learned: $\sqrt{X^2} = |X|$, and I totally understand why. But, when expressed as an exponent, doesn't this really just mean the following: $X^{2/2} = |X|$,…
Christian Castillo
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Spivak's Calculus (Chapter 2, Exercise 17)

I am having trouble completing exercise 17 of chapter 2 of Spivak's Calculus. In this exercise, the reader is asked to prove that for all natural numbers $n$ and $p$, there exist real numbers $\alpha_1,\ldots,\alpha_p$ such that: $$ \sum_{k = 1}^n…
emi
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wrongly asked question about precalculus?

Im seeing the following question in a precalc textbook: Suppose $f$ is a function whose domain is $[-5,5]$ and $f(x) = \frac{x}{x+3}$ for every $x$ in $[0,5]$. Suppose $f$ is an odd function. Evaluate $f(-3)$. Isnt this a bad formulated problem?? I…
user203867
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2 answers

How many positive values of $a$ are possible in $2^{3}\le a\lfloor a\rfloor \le 4^{2} + 1$

How many positive values of $a$ are possible in the following case? $$2^{3}\le a\lfloor a\rfloor \le 4^{2} + 1$$ where $a\lfloor a\rfloor$ such that $a[a]$ is an integer.
TheApe
  • 553
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1 answer

Deriving formula for asymptotes of a hyperbola

I'm trying to find a precalculus-level derivation of the formula for the asymptotes of a hyperbola. My book says: Solving $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ for $y$, we obtain $y = \pm \frac ba \sqrt{x^2 - a^2}$ $ = \pm \frac ba \sqrt{x^2(1 -…
moonman239
  • 553
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3 answers

Solve the equation $\sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$

Solve the following equation: $\sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$ Unfortunately I have no idea.
RFZ
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Why are restrictions important?

When simplifying expressions, why do we add on restrictions for the simplified form if the original form was undefined at a certain point? The simplified form is defined at those points, so why should it be restricted? An example of what I…
user26649
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What is $x^{1/2}$?

I was wondering what $x^{1/2}$ is. You know, when I say $x^2$ it's $x \cdot x$ or $x^3 = x \cdot x \cdot x$ etc. But what is $x ^{1/2}$? I know it's $$\sqrt x$$ but I mean when you want to explain it like with $$x^2 = x \cdot x$$ How do you explain…
Hudhud
  • 61
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If $x>0$ is such that $x^{n}+\frac{1}{x^n}$ and $x^{n+1}+\frac{1}{x^{n+1}}\in \mathbb{Q} \implies x+\frac{1}{x}\in\mathbb{Q}$?

Let $n \in \mathbb{N}$. If $x>0$ is such that $x^{n}+\frac{1}{x^n}$ and $x^{n+1}+\frac{1}{x^{n+1}}\in \mathbb{Q} \implies x+\frac{1}{x}\in\mathbb{Q}$? Any thoughts on how to solve the above problem. Working for $n=2$ says that this result is true,…
Axler
  • 61
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What's $\alpha+\beta$ if we have: $\alpha^3-6\alpha^2+13\alpha=1$ and $\beta^3-6\beta^2+13\beta=19$ ($\alpha$ and $\beta$ are Real)

What's $\alpha+\beta$ if we have $\alpha^3-6\alpha^2+13\alpha=1$ and $\beta^3-6\beta^2+13\beta=19$? Here $\alpha$ and $\beta$ are real. Firstly, I subtracted the two equations and got the…
Hamid Reza Ebrahimi
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Why can't you have more turning points than the degree?

I get that each degree can correspond to a factor. $x^5=(x+a)(x+b)(x+c)(x+d)(x+e)$ and that results in 4 turning points, so the graph can "turn around" and hit the next zero. Why can't a curve have more turning points than zeros? In the…
JackOfAll
  • 4,701
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If $(x,y)$ satisfies $x^2+y^2-4x+2y+1=0$ then the expression $x^2+y^2-10x-6y+34$ CANNOT be equal to

If $(x,y)$ satisfies $x^2+y^2-4x+2y+1=0$ then the expression $x^2+y^2-10x-6y+34$ CANNOT be equal to $(A)\frac{1}{2}\hspace{1cm}(B)8\hspace{1cm}(C)2\hspace{1cm}(D)3$ $(x,y)$ satisfies $x^2+y^2-4x+2y+1=0$ $(x,y)$ satisfies $(x-2)^2+(y+1)^2=2^2$ The…
Brahmagupta
  • 4,204
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The smallest possible value of $x^2 + 4xy + 5y^2 - 4x - 6y + 7$

I have been trying to find the smallest possible value of $x^2 + 4xy + 5y^2 - 4x - 6y + 7$, but I do not seem to have been heading in any direction which is going to give me an answer I feel certain is correct. Any hints on how to algebraically…
user34144
  • 63
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Value of $X^4 + 9x^3+35X^2-X+4$ for $X=-5+2\sqrt{-4}$

Find the value of $X^4 + 9x^3+35X^2-X+4$ for $X=-5+2\sqrt{-4}$ Now the trivial method is to put $X=5+2\sqrt{-4}$ in the polynomial and calculate but this is for $2$ marks only and that takes a hell lot of time for $2$! So I was thinking may be…
user80631
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