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# 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.

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

## 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

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

## Identifying Even and Odd Functions

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.

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

## Comments

Log in to comment## rhill says:

about 3 yearsWhere do I find the "show attached resources" link. Thanks.

## Cam says:

almost 3 yearsOur apologies -- we're working on correcting a bug which removed all of the attached resources. Hopefully they will all be back soon (after which the link should be pretty apparent).

## Cam says:

almost 3 yearsremoved

## Cam says:

almost 3 yearsProblem solved. You can now find the attached resource(s) right above the Solutions pane. Apologes again for the mix-up.

## Marianne Springer says:

almost 4 yearsHow do I access the experimental geogebra worksheet file?

## Kristin says:

almost 4 yearsSorry--we forgot to make it visible. Now it should show up under the "Show attached resources" link.