Signum Function  

The signum function, denoted as sgn(x), is a mathematical function that identifies the sign of a real number. It outputs 1 if the number is positive, -1 if negative, and 0 if the number is zero. This simple yet useful function helps in various fields like calculus, signal processing, and control systems by focusing solely on the sign rather than the magnitude of values. It’s fundamental for defining piecewise and directional behavior.

1.0What is the Signum Function? (Signum Function Definition)

The signum function, often denoted as sgn(x), is a simple function that extracts the sign of a real number. It tells you whether a number is positive, negative, or zero. Instead of focusing on the magnitude of the number, the signum function focuses solely on its sign.

2.0Signum Function Formula and Symbol

The signum function formula is defined as

$\mathrm{sgn}(x)=\{\begin{array}{ll}-1& \text{ if }x<0\\ 0& \text{ if }x=0\\ 1& \text{ if }x>0\end{array}$

The signum function symbol is typically written as sgn(x).

3.0Domain and Range of the Signum Function

• Domain: The signum function is defined for all real numbers, so its domain is

Domain = (−∞, ∞)

• Range: Since the function outputs only three possible values (-1, 0, or 1), the range is

Range = {−1, 0, 1}  

4.0Graph of the Signum Function

The signum function graph is a step-like graph with three distinct parts:

• A flat line at -1 for all negative values of x.
• A single point at 0 when x = 0.
• A flat line at 1 for all positive values of x.

This graph is discontinuous at x = 0, which is where the function value jumps from −1 to 1, passing through 0.

5.0Derivative of the Signum Function

The derivative of the signum function is closely related to the Dirac delta function (δ). Since the signum function jumps abruptly at zero, its derivative is zero everywhere except at x = 0, where it is undefined in the traditional sense.

Mathematically, the derivative can be expressed as

$\frac{d}{dx}\mathrm{sgn}(x)=2\delta (x)$

where δ(x) is the Dirac delta function.

6.0Uses of the Signum Function

The Signum function has a variety of applications, including:

• Signal processing: To detect the sign of a signal or to create piecewise functions.
• Control systems: For designing systems that respond differently to positive and negative inputs.
• Mathematics: In calculus and differential equations, especially for defining piecewise functions and solving problems involving absolute values.
• Computer science: To simplify decision-making algorithms that depend on whether values are positive, negative, or zero.
• Physics: Modeling directional quantities where only the direction (positive or negative) matters.

7.0Solved Examples on Signum Function 

Example 1: Find sgn(−10), sgn(0), and sgn(5).

Solution: 

sgn(−10) = −1 (since −10 < 0) 

sgn(0) = 0 

sgn(5) = 1 (since 5 > 0)  

Example 2: Evaluate $\mathrm{sgn}\left(x^2-9\right)$ for x = –4, 0, 3.

Solution: 

• For x = –4:

$x^2-9=16-9=7>0\Rightarrow \mathrm{sgn}(7)=1$

• For x = 0:

$0^2-9=-9<0\Rightarrow \mathrm{sgn}(-9)=-1$

• For x = 3:

$3^2-9=9-9=0\Rightarrow \mathrm{sgn}(0)=0$

Example 3: Solve for x if sgn(2x−4)=1.

Solution: 

$\begin{array}{rl}& \mathrm{sgn}(2x-4)=1\\ & \Rightarrow 2x-4>0\\ & \Rightarrow 2x>4\\ & \Rightarrow x>2\end{array}$

So, all x > 2 satisfy the equation.

Example 4: Simplify the expression _{f(x)=x \cdot \operatorname{sgn}(x)}.

Solution: 

• If x > 0, sgn(x) = 1, so f(x) = x⋅1 = x.
• If x = 0, f(x) = 0⋅0 = 0.
• If x < 0, sgn(x) = −1, so f(x) = x⋅(−1) = −x.

Therefore,

f(x) = |x| 

Example 5: Differentiate the function f(x) = |x| for x ≠ 0.

Solution: 

Recall,

∣x∣ = x⋅sgn(x)  

Differentiating,

$\begin{array}{rl}& f^{\prime }(x)=\frac{d}{dx}[x\cdot \mathrm{sgn}(x)]\\ & =\mathrm{sgn}(x)+x\cdot \frac{d}{dx}[\mathrm{sgn}(x)]\end{array}$

But since sgn(x) is constant () except at x = 0, the derivative $\frac{d}{dx}[\mathrm{sgn}(x)]=0$ for x ≠ 0.

Thus,

$f^{\prime }(x)=\mathrm{sgn}(x)$

Example 6: Verify if $\mathrm{sgn}(xy)=\mathrm{sgn}(x)\cdot \mathrm{sgn}(y)forx=–3,y=4$.

Solution:
Calculate the left side:

sgn(−3 × 4) = sgn(−12) = −1  

Calculate the right side:

$\mathrm{sgn}(-3)\cdot \mathrm{sgn}(4)=(-1)\times 1=-1$

Since both sides equal −1, the relation holds for these values.

Example 7: What is the signum function?

Ans: The signum function, $\mathrm{sgn}(x)$, returns the sign of a real number x. It is defined as:

$\mathrm{sgn}(x)=\{\begin{array}{ll}-1& x<0\\ 0& x=0\\ 1& x>0\end{array}$

Example 8: How is the signum function related to the absolute value function?

Ans: The absolute value of xx can be expressed as:

$|x|=x\cdot \mathrm{sgn}(x)\text{ for }x\ne 0$ 

and equivalently,

$\mathrm{sgn}(x)=\frac{x}{|x|}\text{ for }x\ne 0$. 

Example 9: What is the derivative of the absolute value function f(x) = |x| using the signum function?

Ans: For x ≠ 0, the derivative is:

$\frac{d}{dx}|x|=\mathrm{sgn}(x)$. 

At x=0, the derivative is undefined.

Example 10: What is the relation between the signum function and the Heaviside step function?

Ans: The Heaviside step function H(x) is related to the signum function by:

$H(x)=\frac{1+\mathrm{sgn}(x)}{2}$. 

Example 11: Is the signum function an odd or even function?

Ans: The signum function is an odd function:

$\mathrm{sgn}(-x)=-\mathrm{sgn}(x)$. 

8.0Practice Questions on Signum function 

1. Evaluate: $\mathrm{sgn}(-7),\mathrm{sgn}(0),\mathrm{sgn}(3.5)$

2. Simplify: $\mathrm{sgn}\left(x^2-4\right)$ For x = –3, 0, 2, 3.

3. Solve for x: $\mathrm{sgn}(2x-5)=1$

4. If $\mathrm{sgn}(x-1)=-1$, what can you say about x?

5. Express the absolute value function |x| in terms of x and $\mathrm{sgn}(x)$.

6. Differentiate the function f(x) = |x| using the signum function.

7. Prove or disprove: $\mathrm{sgn}(xy)=\mathrm{sgn}(x)\cdot \mathrm{sgn}(y)$ for all real numbers x, y.

8. Given$f(x)=x\cdot \mathrm{sgn}(x)$, simplify f(x).

9. For x ≠ 0, show that

$\mathrm{sgn}(x)=\frac{x}{|x|}$

10. Solve the equation:

$\mathrm{sgn}\left(x^2-1\right)+\mathrm{sgn}(x-1)=0$