Precalculus / Unit 1: Functions and Their Graphs
1.3 Even and Odd Functions
Notes
- Even function: $f(-x) = f(x)$. The graph is symmetric about the y-axis. Example: $x^4 + 3x^2$.
- Odd function: $f(-x) = -f(x)$. The graph is symmetric about the origin. Example: $x^3 - x$.
- A function can be neither, like $x^2 + x$.
- Test: replace $x$ with $-x$ and simplify.
- Example: $f(x) = x^3 - x$, so $f(-x) = -x^3 + x = -(x^3 - x) = -f(x)$, which is odd.
Practice
Which function is even?
Only $x^4 + 3x^2$ stays the same when $x$ is replaced by $-x$.
If $f$ is odd and $f(2) = 5$, what is $f(-2)$?
For an odd function, $f(-2) = -f(2) = -5$.
The function $f(x) = x^2 + x$ is:
$f(-x) = x^2 - x$, which is neither $f(x)$ nor $-f(x)$.
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