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Symmetry of Functions

CSCA Symmetry of Functions study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Symmetry of Functions

1. Concept

**Symmetry** describes the mirroring or rotational relationship of a function's graph.

* **Axial Symmetry (Reflection)**: The graph is symmetric about a vertical line $x=a$. (e.g., Even functions correspond to the axis $x=0$).

* **Central Symmetry (Rotation)**: The graph is symmetric about a point $(a, b)$. (e.g., Odd functions correspond to the center $(0,0)$).

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2. Mathematical Formulas

Let the domain of $f(x)$ be $D$.

#### (1) Symmetry about line $x=a$

* **Geometric Meaning**: Any point on the graph has a mirror image across the line $x=a$ at the same height.

* **Formula**:

$$f(a+x) = f(a-x)$$

Or equivalently:

$$f(x) = f(2a-x)$$

* **Special Case**: If $a=0$, $f(x)=f(-x)$ (Even Function).

#### (2) Symmetry about point $(a, b)$

* **Geometric Meaning**: The graph matches itself if rotated $180^{\circ}$ around $(a, b)$.

* **Formula**:

$$f(a+x) + f(a-x) = 2b$$

Or equivalently:

$$f(x) + f(2a-x) = 2b$$

* **Special Case**: If $(a, b)=(0,0)$, $f(x)+f(-x)=0 \Rightarrow f(-x)=-f(x)$ (Odd Function).

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3. Relation to Periodicity (Important Inference)

If a function has **two** distinct symmetries, it implies periodicity.

* **Two Axes**: Symmetric about $x=a$ and $x=b$ $\Rightarrow$ Periodic with $T = 2|a-b|$.

* **Two Centers**: Symmetric about $(a, 0)$ and $(b, 0)$ $\Rightarrow$ Periodic with $T = 2|a-b|$.

* **One Axis + One Center**: Symmetric about $x=a$ and $(b, 0)$ $\Rightarrow$ Periodic with $T = 4|a-b|$.

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4. Typical Examples

**Example 1: Proving Axial Symmetry**

Prove $f(x) = (x-1)^2$ is symmetric about $x=1$.

**Proof**:

Check if $f(1+x) = f(1-x)$.

LHS $= [(1+x)-1]^2 = x^2$.

RHS $= [(1-x)-1]^2 = (-x)^2 = x^2$.

Since LHS = RHS, it is symmetric about $x=1$.

**Example 2: Using Central Symmetry**

$f(x)$ is symmetric about $(1, 2)$, and $f(0)=3$. Find $f(2)$.

**Solution**:

Formula: $f(1+x) + f(1-x) = 2(2) = 4$.

Let $x=1$: $f(2) + f(0) = 4$.

$f(2) + 3 = 4 \Rightarrow f(2) = 1$.

**Example 3: Symmetry to Periodicity**

$f(x)$ is symmetric about lines $x=1$ and $x=2$. Prove it is periodic.

**Proof**:

1. Symmetry about $x=2 \Rightarrow f(x) = f(4-x)$.

2. Symmetry about $x=1 \Rightarrow f(4-x) = f(2-(4-x)) = f(x-2)$.

3. Therefore, $f(x) = f(x-2)$.

This fits the definition of a periodic function with $T=2$.

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5. Common Pitfalls

* **Confusing Formulas**: Axial symmetry implies equality ($f(a+x)=f(a-x)$), while central symmetry involves a sum ($f(a+x)+f(a-x)=2b$).

* **Domain Issues**: Symmetry requires the domain itself to be symmetric about the axis or center.

* **Misinterpreting Periodicity**: Just because $f(1+x) = f(1-x)$ does NOT mean the period is 1 or 2. That is symmetry, not translation.