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Parity of Functions (Even/Odd)

CSCA Parity of Functions (Even/Odd) study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Parity of Functions (Even/Odd)

1. Concept

**Parity** describes the symmetry of a function's graph. It is a global property.

**Core Prerequisite**: The domain $D$ MUST be **symmetric about the origin**. If $x \in D$, then $-x \in D$.

Given the domain is symmetric:

* **Even Function**:

* **Definition**: For all $x \in D$, $f(-x) = f(x)$.

* **Graph**: Symmetric about the **y-axis** (like folding paper along the y-axis).

* **Examples**: $y = x^2, y = \cos x, y = |x|$.

* **Odd Function**:

* **Definition**: For all $x \in D$, $f(-x) = -f(x)$.

* **Graph**: Symmetric about the **origin $(0,0)$** (rotational symmetry of $180^\circ$).

* **Examples**: $y = x^3, y = \sin x, y = 1/x$.

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2. Methods of Determination

#### (1) Definition Method (Standard)

1. **Check Domain**: Is it symmetric about the origin? If NO $\rightarrow$ **Neither even nor odd**.

2. **Compute $f(-x)$**: Substitute $-x$ and simplify.

3. **Compare**:

* $f(-x) = f(x) \rightarrow$ **Even**

* $f(-x) = -f(x) \rightarrow$ **Odd**

* Neither $\rightarrow$ **Neither**

#### (2) Operational Properties (Quick Check)

* **Odd $\pm$ Odd = Odd**

* **Even $\pm$ Even = Even**

* **Odd $\times$ Odd = Even**

* **Even $\times$ Even = Even**

* **Odd $\times$ Even = Odd**

* **Composition**: If inner is even, composite is even. Inner odd + Outer even = Even. Inner odd + Outer odd = Odd.

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3. Typical Examples

**Example 1: Definition Method**

Determine the parity of $f(x) = x^2 + \cos x$.

**Solution**:

1. **Domain**: $\mathbb{R}$, symmetric.

2. **Compute**: $f(-x) = (-x)^2 + \cos(-x)$.

Since $(-x)^2 = x^2$ and $\cos(-x) = \cos x$.

$f(-x) = x^2 + \cos x = f(x)$.

**Result**: **Even Function**.

**Example 2: The Domain Trap**

Determine the parity of $f(x) = \sqrt{x}$.

**Solution**:

1. **Domain**: $x \ge 0$, i.e., $[0, +\infty)$.

2. **Check**: Domain is NOT symmetric (e.g., $1$ is in, $-1$ is not).

**Result**: **Neither even nor odd**.

**Example 3: Applying Properties**

$f(x)$ is an odd function on $\mathbb{R}$. For $x>0$, $f(x)=x^2-2x$. Find $f(-1)$.

**Solution**:

Property of odd function: $f(-1) = -f(1)$.

Calculate $f(1) = 1^2 - 2(1) = -1$.

So $f(-1) = -(-1) = 1$.

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4. Common Pitfalls

* **Ignoring Domain**: This is a common mistake. For $f(x) = x^2 + \sqrt{x-1}$, calculating $f(-x)$ is wrong because the domain $x \ge 1$ is not symmetric.

* **The Zero Property**: If an odd function is defined at $x=0$, then $f(0)$ MUST be $0$. This is often a key to solving problems.

* **Both Even and Odd**: The function $f(x) = 0$ (with a symmetric domain) is the only function that is both even and odd.