# Functions

## Function Transformations as Composition

(A new question of the week) We have discussed transformations of functions and their graphs at length, but a recent question suggested a slightly different way to think about them.

## Why Are Functions Defined as They Are?

Last week we looked at what functions are; but many students wonder why it all matters. What makes them useful? What makes functions worth distinguishing from non-functions? Why do we make the distinction we do? We love “why” questions, because they make us think more deeply!

## What is a Function, Really?

A recent question, from Anindita, touched on the relationship of functions, relations, and rules. I referred to several answers we’ve given, which I’d long planned to put into a post (or two). This is it! We’ll start with a set of questions about what functions are.

## 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 …

## Average Rate of Change of a Function

(A new question of the week) Average rate of change is a topic taught in pre-calculus and calculus courses, primarily as preparation for the derivative, though it has more immediate applications. A recent question asked about when the concept is valid, which I found interesting.

## Un-piecing and Inverting a Piecewise Function

(A new question of the week) Though students often think piecewise-defined functions are unnatural, they are actually quite common in real life – after all, the world is not all once piece! In particular, they show up in various financial situations, such as taxes and pay rates. Inverses, likewise, are commonly needed in real life, …

## Domain and Range of a Radical Function

(A new question of the week) We’ve looked at domain and range problems before, but some have more interesting details than others. Here is a superficially basic radical function (and the answer is extremely easy when you just use a graphing tool), which raised some interesting issues while solving it algebraically.

## Exponential Growth

The term “exponential” has gone viral, so to speak. Do we all know what it means? In the next few posts I’ll look at answers we’ve given to questions about exponential growth and related concepts, some of them about the spread of diseases or rumors. (Disclaimer: I will be writing about the basic math, not …

## Domain, Range, and Quadratic Inequalities

(A new question of the week) We had a long discussion last August about domain and range of functions involving either quadratic functions or restricted domains (or both). Two Math Doctors got involved, offering different ways to approach the same problem. I’ve edited the discussion to avoid some intermingling of topics. A quadratic inequality Here …

## Equivocal Function Transformations

The last two posts were about transformations of functions (shift, stretch, reflect) and their effect on a graph, first individually and then in combination. The next thing to look at will be how to determine the transformations when you are given a graph; but before we take that challenge in general, we need to see …