# Functions

## 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 …

## What Are Trig Functions, Really?

(An archive question of the week) Trigonometric functions are sometimes introduced without a deep explanation of their meaning; they are just buttons to push on a calculator, or names to write in an equation. Even when a textbook gives a careful presentation, there are so many facets to the concept that it can be easy …

## Ranges of Inverse Trig Functions

(Archive Question of the Week) We have had a number of questions over the years about inverse trig functions and their ranges. For today’s question, I have chosen one from 2011, which will link to a number of others that I will not quote in detail.

## Finding the Range of a Function

Recall that the domain of a function is the set of all valid input values (x), and the range is the set of all possible output values (y). It is reasonably easy to find the domain: look for what could make it impossible to evaluate, such as dividing by zero or taking the square root …