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Exercises Functions

Example 12 from Algebra and Trigonometry, 3.5 Transformation of Functions

Determining whether a Function Is Even, Odd, or Neither

Is the function $f (x) = x^{3} + 2 x$ even, odd, or neither?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let’s begin with the rule for even functions.

$f (- x) = {(- x)}^{3} + 2 (- x) = - x^{3} - 2 x$

This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.

$- f (- x) = - (- x^{3} - 2 x) = x^{3} + 2 x$

Because $- f (- x) = f (x),$ this is an odd function.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Is the function $f (s) = s^{4} + 3 s^{2} + 7$ even, odd, or neither?

How did it go?

PreviousApplying a Learning Model Equation Next Graphing a Vertical Stretch

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 3.5, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.