Identifying Even and Odd Functions

Task

Determine whether each of these functions is odd, even, or neither. Use algebraic methods on all of the functions. You may start out by looking at a graph, if you need to.

1. $f(x) = 3^x + 3^{-x}$
2. $g(x) = 2^x – 2^{-x}$
3. $h(x) = x^2 + 4x - 2$
4. $j(x) = x^3 – 4x$

IM Commentary

This task includes an experimental GeoGebra worksheet, with the intent that instructors might use it to more interactively demonstrate the relevant content material. The file should be considered a draft version, and feedback on it in the comment section is highly encouraged, both in terms of suggestions for improvement and for ideas on using it effectively. The file can be run via the free online application GeoGebra, or run locally if GeoGebra has been installed on a computer.

Attached Resources

Solution

1. $f(x)$ is even since $f(-x) = 3^{-x} + 3^x = f(x)$.
2. $g(x)$ is odd since $g(-x) = 2^{-x} – 2^x = -g(x)$.
3. $h(x)$ is neither odd nor even since $h(-x)$ equals neither $h(x)$ nor $-h(x)$.
4. $j(x)$ is also odd, since $j(-x)=(-x)^3-4(-x) = -(x^3)+4x=-j(x)$.