Articles of algebra precalculus

Describing asymptotic behaviour of a function

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution

How do I solve this equation involving a logarithm?

I’m running in circles and I don’t understand how to do this. $$x\log(x) = 100$$ Where the $\log$ is in base $10$, I understand that $\log(y)=x$ is $10^x = y$. So is it the same for $x\log(x) = 100$? Would it be $10^{100}=x\cdot x$? It doesn’t come out right when I do it, and it’s […]

Number of solutions of $a^{3}+2^{a+1}=a^4$.

Find the number of solutions of the following equation $$a^{3}+2^{a+1}=a^4,\ \ 1\leq a\leq 99,\ \ a\in\mathbb{N}$$. I tried , $$a^{3}+2^{a+1}=a^4\\ 2^{a+1}=a^4-a^{3}\\ 2^{a+1}=a^{3}(a-1)\\ (a+1)\log 2=3\log a+\log (a-1)\\ $$ This is from chapter quadratic equations. I look for a short and simple way. I have studied maths up to $12$th grade.

How can we write a piecewise function in the form $u(t-a)f(t-a)$?

Given a piecewise function, such as $$f(t) = \begin{cases} 2, & \text{if }t \lt a \\ t^2, & \text{if }t \geq a \end{cases}$$ Or some other piecewise function, how can we write it in the form $u(t-a)f(t-a)$ for $$u(t-a) = \begin{cases} 0, & \text{if }t \lt a \\ 1, & \text{if }t \geq a \end{cases}$$ […]

Calculate the height of a building

This question I really need help with, I simply do not know where to start! Anyone can help, all I can offer is supreme thanks. Please include method. I don’t want simple answers which don’t help me learn

Inequalities – Absolute Value $|2x-1| \leq |x-3|$

$$|2x-1| \leq |x-3|$$ Answer is $$-2 \leq x \leq \frac43$$ My Question is HOW?

$n^s=(n)_s+f(s)$, what is $f(s)$?

In the following equation, $$n^s=(n)_s+f(s)$$ What is general form for $f(s)$? Understand that, $$(n)_s=n(n-1)(n-2)\cdots(n-[s-1])=\text{ The Falling Factorial }$$ I have experimented with this equation for $s=\{1,2,3,4\}$. Unless my calculations are horribly mangled, the following table arises: $$ \begin{array}{c|c} s & f(s)\\ \hline 1 & 0\\ 2 & n\\ 3 & 3n^2-2n\\ 4 & 6n^3-11n^2+6n \end{array} […]

Changing the cartesian coordinate system

The cartesian coordinate in 3D is given as: Are we allowed to make our own coordinate system (switching axes around). The question is can we change the axes around? ** Like: **LOOKING AT THE FIRST ONE: Change the $z$-axis to the $x$-axis? or change the $y$-axis to the $z$-axis?? Also in 2D for example: Can […]

no. of Digit in $x^y\;,$ where $x,y\in \mathbb{N}$

$(1)$:: Calculation of no. of Digits in $2^{100}$ .$(2)$:: Calculation of no. of Digits in $3^{100}$. If it is given that $\log_{10}(2)=0.3010$ and $\log_{10}(3) = 0.4771$ $\bf{My\; Try::}$ I have seen in book and it is given as :: $(1)$ no. of Digit in $\displaystyle 2^{100}$ is equal to $\displaystyle \lfloor \log_{10}(2)^{100}\rfloor +1\;,$ where $\lfloor […]

How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on their own on the books given on the answer to that question Also, i think it is too much […]