Two Worlds of Relations
Terminology and definitions in mathematics sometimes vary according to context. Here we’ll look at the ideas of relations, functions, and their domains, and discover that they look different from different perspectives.
Algebra
Function Transformations Revisited (II)
(A new question of the week) Last week we examined how a series of transformations affects the equation of a function, in order to write the equation from a graph, or vice versa. We touched on why it works the way it does, but this is something you need to look at from multiple perspectives …
Function Transformations Revisited (I)
(A new question of the week) Transformations of functions, which we covered in January 2019 with a series of posts, is a frequent topic, which can be explained in a number of different ways. A recent discussion brought out some approaches that nicely supplement what we have said before. Here, the focus will be on …
Why Does a² + b² = c² in a Hyperbola?
(A new question of the week) In an ellipse, \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) with focal distance c, parameters a, b, and c all make natural sense, and it is easy enough to see why \(a^2 = b^2 + c^2\). But in the hyperbola, \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), the equivalent relationship, \(a^2 + b^2 = c^2\), is not nearly as natural, nor …
Trigonometry, Radicals, and a Very Common Error
(A new question of the week) While I was looking through recent questions to choose one to post, I ran across one that deals with an error we see very commonly – in fact, a student I had worked with that very afternoon in face-to-face tutoring had done the same sort of thing. The context …
Trying to Solve a Strange Log Equation
(A new question of the week) We’ll look at a very complicated logarithmic equation, which leads to quartic equations and some very interesting graphs. We won’t find a fully satisfying solution method, but we’ll have some fun trying – and reveal the fallibility of at least one Math Doctor!
Average Distance Between Two Sets of Points
(A new question of the week) Here we have a different kind of question than usual: A conjecture about distances between points, with a request for confirmation. Normally we like to just give hints to help a student figure something out; this was a request for a theorem that ought to exist, and trying to …
Two Word Problems About Factors and Sums
(A new question of the week) A couple recent questions involved factoring numbers, in interesting ways. One involves the volume and perimeter of a block of cubes, and the other involves finding numbers with a given HCF (Highest Common Factor) and sum. Both illustrate thinking through a non-routine problem about factors.
Multiplication, Division, and Powers of Ten
(A new question of the week) We’ve looked in the past at place values, but here we’ll see some tricks for doing multiplication and division with both decimals and large numbers by moving the decimal point around. The first question is primarily a matter of arithmetic, then the second extends it to the algebraic concept …
Finding a Function Value Recursively
(A new question of the week) May was a particularly good month for interesting questions! Here is one requiring us to find one value of a function, based on an unusual property: If \(a+b=2^x\), then \(f(a)+f(b)=x^2\). The problem turned out to be not as hard as it looked, yet the function itself is quite interesting …