Articles of algebra precalculus

Is parametric form of a given function unique?

Can we say that for any given function in single/multivariable, it is always possible to have a parametric form? (Elementary functions, complicated functions?) Given any function, is parametric form uniquely determined?

How can $r$ be negative when dealing with polar coordinates?

If by definition $r=\sqrt{x^2 + y^2}$, then why do we allow $r$ to be negative? Relatedly, I do not understand the last section of this conversation discussing points being represented by multiple $\theta$: Student: So a single point could have many different values? Mentor: Correct! The values for $r$ can be given as positive and […]

Find $r$ knowing that $r=\frac{60}{\sin^{-1}\frac{60}{r}}$

I’m trying to find the value of $r$ knowing that: $$r=\frac{60}{\sin^{-1}\frac{60}{r}}$$ I’m not really sure how to approach finding the solution. Can anyone help me out? I’ve spent well over an hour on the problem to get to this point, and now I’m stuck. Thanks!

On Profit and loss

A dishonest dealer marks his goods $20\%$ above the cost price. He gives a discount of $10\%$ to the customer on the marked price and makes a profit by using a false weight of $900$ gms in place of $1$ kg while buying or selling. Find the percentage profit earned by the shopkeeper. I do […]

Sums of two perfect squares

Show that if $q$ is a number that can be expressed as the sum of two perfect squares, then $2q$ and $5q$ can also be expressed as the sum of two perfect squares. EDIT: I’ve recently revisited this problem and I found an elementary answer which I posted as an answer below.

Superball total bounce distance

I am asked to explain how to calculate total bounce distance: A “super” rubber ball is one that is measured to bounce when dropped 70% or higher that the distance from which it is dropped. You are to take a super rubber ball that bounces 75% of it dropped height and you are to find […]

Given $r>0$, find $k>0$ such that $\sqrt{(x-2)^2+(y-1)^2}<k$ implies $|xy-2|<r $

Using the axioms, theorem, definitions of high school algebra concerning the real numbers, then prove the following: Given $r>0$, find a $k>0$ such that: $$\text{for all }x, y: \sqrt{(x-2)^2+(y-1)^2}<k\implies|xy-2|<r $$ I tried with several values given to $k$ and $r$ to find the relation between them. Suppose then $r=1$ and we choose $k$ to be […]

How do you prove a function is defined for a a certain set?

Spin-off from here. Context: Highschool textbooks often ask students to find the domain of functions. Let’s say $f(x) = x+2$. The domain is $\mathbb{R}$…suppose a student (highschool or o/w) asks why $f(x)$ is defined ∀$\mathbb{R}$…? What about $f(x)=5e^{x}$? $f(x)=\ln(x)$ for $\mathbb{R}^{+}$?

Evaluating $\sum_{k=1}^n k^x$

Possible Duplicate: Finite Sum of Power? Is there a general expression for $\sum_{k=1}^n k^x$ for any integer value of $x$? The table for $x=1,2,\dots 10$ is given here. Is there formula for any value of $x$?

$x^2+y^2+9=3(x+y)+xy$ Find all pairs of real $x,y$ that meet this equation

$\frac{(x-y)^2}{(y-3)(3-x)} = 1$ That was my attempt, I can’t think of anything else here. I’d prefer a hint