Function Notation: Even and Odd Functions


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Function Notation: Even and Odd (page 3 of 3)

Sections: Introduction & Evaluating at a number, Evaluating at a variable, Even and odd functions

You may be asked to determine algebraically whether a function is even or odd. To do this, you take the function and plug –x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f(–x) = f(x)), then the function is even. If you end up with the exact opposite of what you started with (that is, if f(–x) = –f(x)), then the function is odd. In all other cases, the function is neither even nor odd. Here are some examples:

Determine algebraically whether f(x) = –3x² + 4 is even, odd, or neither.

If I graph this, I will see that this is "symmetric about the y-axis"; in other words, whatever the graph is doing on one side of the y-axis is mirrored on the other side:

This mirroring about the axis is a hallmark of even functions.

Note also that all the exponents are even (the exponent on the constant term being zero: 4x⁰ = 4 ×1 = 4).

But the question asks me to make the determination algebraically, which means that I need to do it with algebra, not with graphs.

graph of –3x^2 + 4

So I'll plug –x in for x, and simplify:

f(–x) = –3(–x)² + 4
= –3(x²) + 4
= –3x² + 4

This is the same thing I started with. That means that f(x) is even.

Determine algebraically whether f(x) = 2x³ – 4x is even, odd, or neither.

If I graph this, I will see that it is "symmetric about the origin"; that is, if I start at a point on the graph on one side of the y-axis, and draw a line from that point through the origin and extending the same length on the other side of the y-axis, I will get to another point on the graph.

This symmetry is a hallmark of odd functions.

Note also that all the exponents are odd (since the second term is 4x = 4x¹). This is a useful clue. I would expect this function to be odd.

graph of 2x^3 – 4x

But the question asks me to make the determination algebraically, so I'll plug –x in for x, and simplify:

f(–x) = 2(–x)³ – 4(–x)
= 2(–x³) + 4x
= –2x³ + 4x

This is the exact opposite of what I started with:

–f(x) = –[2x³ – 4x]
= –2x³ + 4x

That means that f(x) is odd.

Determine algebraically whether f(x) = 2x³ – 3x² – 4x + 4 is even, odd, or neither.

This function is the sum of the previous two functions.

Note that its graph does not have the symmetry of either of the previous ones. nor are all its exponents either even or odd.

I would expect this function to be neither even nor odd.

I"ll plug –x in for x, and simplify:

graph of 2x^3 – 3x^2 – 4x + 4

f(–x) = 2(–x)³ – 3(–x)² – 4(–x) + 4
= 2(–x³) – 3(x²) + 4x + 4
= –2x³ –3x² + 4x + 4

This is neither the same thing I started with (2x³ – 3x² – 4x + 4) nor the exact opposite of what I started with (–2x³ + 3x² + 4x – 4). This means that

f(x) is neither even nor odd.

You may find it helpful, when answering this "even or odd" type of question, to write down f(x) and –f(x) explicitly, and compare them to f(–x). This can help you make a sure determination of the correct answer.

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Cite this article as:

Stapel, Elizabeth. "Function Notation: Even and Odd." Purplemath. Available from
http://www.purplemath.com/modules/fcnnot3.htm. Accessed

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