# Functions

## How to Think About the Chain Rule

Having recently helped some students (in person) with the rules of differentiation, I’m reminded to do so here, starting with the chain rule. It is easy to make this topic look harder than it really is; the two main ways to state the rule are often confusing, and different approaches fit different problems. We’ll try …

## More On Mixing Trig Functions

I’ve had several occasions in face-to-face tutoring lately to refer to a past post on mixing (that is, composition) of trig and inverse trig functions. Several recent questions have touched directly or indirectly on this same general idea and extended it, so I thought I’d post them.

## Monotonic Functions, Inequalities, and Optimization

Looking for a cluster of questions on similar topics, I found several from this year in which monotonic functions (functions that either always increase, or always decrease) provide shortcuts for various types of problems (optimization with or without calculus, and also algebraic inequalities). We’ll look at a few of these.

## 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.