Articles of algebra precalculus

Why $x^{(1/2)2} \neq x^{2(1/2)} $?

I know, probably is a newb. question, but i can’t get this $x^{(1/2)2} \neq x^{2(1/2)} $ $ x\in\mathbb R^+$. I know $x^{(1/2)2}=(\pm \sqrt{x})^2=+x $ and $x^{2(1/2)}=\pm x $ because $(+x)^2=(-x)^2=x^2$. Edit so $\pm x \neq +x$ so…maybe multiplication is not commutative and $(1/2)2 \neq 2(1/2)$, or maybe $\sqrt[n]{x}\neq x^{1/n}$… or $(x^n)^m\neq x^{n\times m} $ in […]

Alternate Proof for one-one function

I need an alternate proof for this problem. Show that the function is one-one, provide a proof. $f:x \rightarrow x^3 + x : x \in \mathbb{R}$ I needed to show that the function is a one-one function. I tried doing $f(x) = f(y) \Rightarrow x = y$, It ended up with $x(x^2 + 1) = […]

A “fast” and “manual” approach for sorting a set of fractions

What could be the fastest manual approach for sorting (ascending) these fractions: $$\frac{117}{229},\frac{143}{291},\frac{123}{241},\frac{109}{219}$$ I would also be very grateful if somebody explain a general manual approach for sorting fractions which doesn’t really follow any pattern.

System of Equations: any solutions at all?

I am looking for any complex number solutions to the system of equations: $$\begin{align} |a|^2+|b|^2+|c|^2&=\frac13 \\ \bar{a}b+a\bar{c}+\bar{b}c&=\frac16 (2+\sqrt{3}i). \end{align}$$ Note I put inequality in the tags as I imagine it is an inequality that shows that this has no solutions (as I suspect is the case). This is connected to my other question… I have […]

Range of a 1-2 function

$$f(x)=\frac x {x^2+1}$$ I want to find range of $f(x)$ and I do like below . If someone has different Idea please Hint me . Thanks in advanced . This is 1-1 function $\\f(x)=\dfrac{ax+b}{cx+d}\\$, This is 2-2 function $\\f(x)=\dfrac{ax^2+bx+c}{a’x^2+b’x+c’}\\$, This is 1-2 function $\\f(x)=\dfrac{ax+b}{a’x^2+b’x+c’}\\$

Drawing sine and cosine waves

I like mathematics and pretty much every mathematical subject, but if there is one thing I thoroughly dislike, it is drawing (functions, waves, diagrams, etc.) We have this important trig test coming up and I need to master the drawing of sine and cosine waves. Can you guys give me an action plan of how […]

When are we permitted to multiply or divide both sides of an equation by a variable?

As it is said in the mathematics books (at least the one I have), we are not permitted to divide or multiply both sides of an equation by a variable, because it is possible to lose some answers. For example, in the following equation $$x^2=x$$ if we divide both sides by $x$, we would have […]

Finding the range of a $y=-x^2(x+5)(x-3)$ without calculus?

I was helping a precalculus student with this question. The graph wasn’t given. My only idea was to find the inverse and try to find its domain. When trying to find the inverse, I arrived at $0=y^4+2y^3-15y^2+x$ and I was hoping to be able to factor it. However, besides using the quartic formula, I’m not […]

Need to simplify ratio test: $\frac{(2(k+1))!}{(2k)!} $

So I am going to check if this series converges. $$\sum_{k=1}^\infty \frac{(2k)!}{k^{k}} $$ I used the ratio test for this, and I end up with this limit: $$\lim_{k \to \infty} \frac{(2(k+1))!\cdot{k^{k}}}{(2k)!\cdot{(k+1)}^{k+1}}$$ I have no clue on how I simplify: $$\frac{(2(k+1))!}{(2k)!} $$ I know for instance: $$\frac{(k+1)!}{k!} = k+1 $$ But the constant 2 infront of […]

Help demonstrate how to arrive at the implication of some given inequalities and equations

Given: $0<x<y<1$ $z=x+y$ $x=u$ $y=z-u$ $0<u<z-u<1$ I need to show that this implies: If $0<z<1$, then $0<u<\frac{z}{2}$, and If $1<z<2$, then $z-1<u<\frac{z}{2}$. I can observe that $0<z<2$ but I am not able to make any progress on getting the 2 required implications. Please show me the steps as well as the reasoning to solve these […]