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College Algebra

Even and Odd Functions

Lesson

Some functions have a built-in symmetry. Spotting it tells you about the graph instantly — without ever plotting a point.

Even function

f(−x)=f(x)f(-x) = f(x)

Plugging in −x-x gives the same value as xx. Graph is symmetric across the y-axis. Examples: x2, x4, ∣x∣, cos⁡xx^2, \ x^4, \ |x|, \ \cos x.

Odd function

f(−x)=−f(x)f(-x) = -f(x)

Plugging in −x-x gives the opposite of f(x)f(x). Graph is symmetric about the origin (a 180° rotation maps it onto itself). Examples: x, x3, x5, sin⁡x, 1/xx, \ x^3, \ x^5, \ \sin x, \ 1/x.

Neither

If f(−x)f(-x) isn’t f(x)f(x) and isn’t −f(x)-f(x), the function is neither. Adding a constant to an odd function (like x3+2x^3 + 2) usually breaks both symmetries.

Worked example 1

f(x)=x2+1f(x) = x^2 + 1
f(−x)=(−x)2+1=x2+1=f(x)f(-x) = (-x)^2 + 1 = x^2 + 1 = f(x)

Same — so ff is even.

Worked example 2

f(x)=x3−xf(x) = x^3 - x
f(−x)=(−x)3−(−x)=−x3+x=−(x3−x)=−f(x)f(-x) = (-x)^3 - (-x) = -x^3 + x = -(x^3 - x) = -f(x)

Opposite — so ff is odd.

Worked example 3

f(x)=x2+xf(x) = x^2 + x
f(−x)=x2−xf(-x) = x^2 - x

Neither equals f(x)f(x) nor −f(x)-f(x), so neither.

How to type your answer

  • Type 1 for EVEN.
  • Type 2 for ODD.
  • Type 0 for NEITHER.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

f(x)=x2f(x) = x^2

Problem 2

f(x)=x3f(x) = x^3

Problem 3

f(x)=xf(x) = x

Problem 4

f(x)=∣x∣f(x) = |x|

Practice

Standard problems matching the lesson.

Problem 5

f(x)=x4f(x) = x^4

Problem 6

f(x)=x5f(x) = x^5

Problem 7

f(x)=x2+1f(x) = x^2 + 1

Problem 8

f(x)=x3−xf(x) = x^3 - x

Problem 9

f(x)=x2+xf(x) = x^2 + x

Problem 10

f(x)=1xf(x) = \frac{1}{x}

Problem 11

f(x)=1x2f(x) = \frac{1}{x^2}

Problem 12

f(x)=x3+2f(x) = x^3 + 2

Problem 13

f(x)=−xf(x) = -x

Problem 14

f(x)=−x2f(x) = -x^2

Problem 15

f(x)=5f(x) = 5

Problem 16

f(x)=x2−4f(x) = x^2 - 4

Problem 17

f(x)=x3+xf(x) = x^3 + x

Problem 18

f(x)=x+1f(x) = x + 1

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

f(x)=2x3−5xf(x) = 2x^3 - 5x

Problem 20

f(x)=3x4+4x2f(x) = 3x^4 + 4x^2

Problem 21

f(x)=x2+x3f(x) = x^2 + x^3

Problem 22

f(x)=(x+1)2f(x) = (x + 1)^2

Problem 23

f(x)=cos⁡xf(x) = \cos x

Problem 24

f(x)=sin⁡xf(x) = \sin x

Problem 25

f(x)=x6−x2f(x) = x^6 - x^2

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