奇偶性qī'ǒuxìng

Core Concept

Parity describes the symmetry properties of a function. A function can be even, odd, or neither.

Prerequisite: For a function to have parity, its domain must be symmetric about the origin (if $x$ is in the domain, then $-x$ must also be in the domain).

Definitions

Even Function (偶函数)

A function $f$ is even if: $f(-x) = f(x) \quad \text{for all } x \text{ in the domain}$

Graphical property: The graph is symmetric about the y-axis.

Odd Function (奇函数)

A function $f$ is odd if: $f(-x) = -f(x) \quad \text{for all } x \text{ in the domain}$

Graphical property: The graph is symmetric about the origin.

Special Property of Odd Functions

If $f$ is odd and $0$ is in the domain, then $f(0) = 0$.

Proof: $f(0) = f(-0) = -f(0)$, so $2f(0) = 0$, thus $f(0) = 0$.

Common Functions and Their Parity

Function	Parity	Verification
$y = x^n$ ($n$ even)	Even	$(-x)^n = x^n$
$y = x^n$ ($n$ odd)	Odd	$(-x)^n = -x^n$
$y = \lvert x \rvert$	Even	$\lvert -x \rvert = \lvert x \rvert$
$y = \sin x$	Odd	$\sin(-x) = -\sin x$
$y = \cos x$	Even	$\cos(-x) = \cos x$
$y = \tan x$	Odd	$\tan(-x) = -\tan x$
$y = a^x$	Neither	$a^{-x} \neq a^x$ and $a^{-x} \neq -a^x$
$y = \log_a x$	Neither	Domain not symmetric

Methods to Determine Parity

Step-by-Step Process

1. Check domain symmetry: Is $-x$ in domain whenever $x$ is?
2. Calculate $f(-x)$: Substitute $-x$ into the function
3. Compare with $f(x)$ and $-f(x)$:
   • If $f(-x) = f(x)$ → Even
   • If $f(-x) = -f(x)$ → Odd
   • Otherwise → Neither

Example 1: Even Function

Determine the parity of $f(x) = x^4 - 2x^2 + 1$.

Step 1: Domain is $\mathbb{R}$, which is symmetric about origin. ✓

Step 2: $f(-x) = (-x)^4 - 2(-x)^2 + 1 = x^4 - 2x^2 + 1 = f(x)$

Conclusion: $f$ is even.

Example 2: Odd Function

Determine the parity of $f(x) = \dfrac{x^3}{x^2 + 1}$.

Step 1: Domain is $\mathbb{R}$, symmetric about origin. ✓

Step 2: $f(-x) = \dfrac{(-x)^3}{(-x)^2 + 1} = \dfrac{-x^3}{x^2 + 1} = -\dfrac{x^3}{x^2 + 1} = -f(x)$

Conclusion: $f$ is odd.

Example 3: Neither

Determine the parity of $f(x) = x + 1$.

Step 1: Domain is $\mathbb{R}$, symmetric. ✓

Step 2: $f(-x) = -x + 1$ $f(x) = x + 1$ $-f(x) = -x - 1$

Since $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$:

Conclusion: $f$ is neither even nor odd.

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus.

Example 1: Basic (Difficulty ★★☆☆☆)

Determine the parity of $f(x) = x^3 - x$.

Solution: $f(-x) = (-x)^3 - (-x) = -x^3 + x = -(x^3 - x) = -f(x)$

Answer: Odd function

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Example 2: Intermediate (Difficulty ★★★☆☆)

If $f(x)$ is an odd function and $f(2) = 3$, find $f(-2) + f(0)$.

Solution:

Since $f$ is odd:

• $f(-2) = -f(2) = -3$
• $f(0) = 0$ (property of odd functions)

$f(-2) + f(0) = -3 + 0 = -3$

Answer: $-3$

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Example 3: Advanced (Difficulty ★★★★☆)

If $f(x) = ax^3 + bx^2 + cx + d$ is an odd function, find the values of $b$ and $d$.

Solution:

For odd function: $f(-x) = -f(x)$

$f(-x) = a(-x)^3 + b(-x)^2 + c(-x) + d = -ax^3 + bx^2 - cx + d$

$-f(x) = -ax^3 - bx^2 - cx - d$

Comparing: $-ax^3 + bx^2 - cx + d = -ax^3 - bx^2 - cx - d$

This requires: $bx^2 = -bx^2$ and $d = -d$

Therefore: $b = 0$ and $d = 0$

Answer: $b = 0$, $d = 0$

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Example 4: Advanced (Difficulty ★★★★☆)

If $f(x) = \ln\left(\sqrt{x^2+1} + x\right)$, prove that $f(x)$ is odd.

Solution:

$f(-x) = \ln\left(\sqrt{(-x)^2+1} + (-x)\right) = \ln\left(\sqrt{x^2+1} - x\right)$

Now compute $f(x) + f(-x)$: $f(x) + f(-x) = \ln\left(\sqrt{x^2+1} + x\right) + \ln\left(\sqrt{x^2+1} - x\right)$

$= \ln\left[\left(\sqrt{x^2+1} + x\right)\left(\sqrt{x^2+1} - x\right)\right]$

$= \ln\left[(x^2+1) - x^2\right] = \ln(1) = 0$

Therefore: $f(-x) = -f(x)$

Conclusion: $f(x)$ is odd.

Parity and Operations

Sum of Functions

$f$	$g$	$f + g$
Even	Even	Even
Odd	Odd	Odd
Even	Odd	Neither (generally)

Product of Functions

$f$	$g$	$f \cdot g$
Even	Even	Even
Odd	Odd	Even
Even	Odd	Odd

Common Mistakes

❌ Mistake 1: Ignoring Domain Symmetry

Wrong: $f(x) = \sqrt{x}$ is even because $\sqrt{x} \geq 0$ ✗

Correct: Domain $[0, +\infty)$ is not symmetric about origin, so parity is undefined. ✓

❌ Mistake 2: Forgetting $f(0) = 0$ for Odd Functions

If $f$ is odd and defined at $x = 0$, then $f(0) = 0$.

❌ Mistake 3: Only Checking One Value

Wrong: $f(1) = f(-1)$, so $f$ is even. ✗

Correct: Must verify $f(-x) = f(x)$ for ALL $x$ in domain. ✓

Study Tips

1. ✅ Check domain first: Symmetric about origin is required
2. ✅ Use algebraic verification: Don't rely on graphs alone
3. ✅ Remember special property: Odd functions pass through origin
4. ✅ Know product rules: odd × odd = even

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💡 Exam Tip: For polynomials, odd functions have only odd-degree terms, even functions have only even-degree terms!