This question was previously asked in

DSSSB TGT Maths Male Subject Concerned - 23 Sep 2018 Shift 2

Option 1 : 4α

**Concept:**

A function is said to be periodic if there exists a positive real number “T” such that,

**f(x + T) = f(x)** for all x ∈ D

where “D” is the domain of the function f(x). The least positive real number “T” (T > 0) is known as the **fundamental period** or simply the **period of the function**. The “T” is not a unique positive number. **All integral multiple of “T” within the domain of the function is also the period of the function**.

**Hence,**

**f(x + n.T) = f(x)**; n ∈ Z, for all x ∈ D

Given:

f(x + 2α) + f(x) = 0 for all** x ∈ ℝ**

**Calculation:**

The given equation can be re-written as:

f(x + 2α) = -f(x) for all **x ∈ ℝ**

Here, our objective is to convert the RHS of the equation as f(x). For this, we need to substitute "x" such that the RHS function acquires the RHS function form. So, **replacing "x" by "x + 2α"**, we have,

⇒ f(x + 2α + 2α) = -f(x + 2α) for all x ∈ ℝ

⇒ f(x + 4α) = -f(x + 2α) for all x ∈ ℝ

⇒ f(x + 4α) = -f(x) for all x ∈ ℝ (∵ x was replaced by x + 2α)

It means **"4α"** is the period of f(x).

**Hence, **

**4α is the period of f(x).**