Relation between graph and types of function

Relation Between Graph and Types of Functions

Functions can be studied effectively using their graphical representations. The graph of a function provides visual insights into its domain, codomain, continuity, and whether the function is injective (one-one), surjective (onto), or bijective. Here, we examine three fundamental functions: the modulus function, the greatest integer function, and the signum function.

1. Modulus Function: ( f(x) = |x| )

Domain: ℝ (all real numbers)
Codomain: ℝ
Range: [0, ∞)

Graph: The curve resembles an upward-opening "V" with its lowest point, or vertex, located at the origin (0, 0). For x ≥ 0, the function behaves like y = x; for x < 0, it behaves like y = –x.

Nature:
- Not injective: The function assigns the same output to different inputs, such as f(2) = 2 and f(−2) = 2.
- Not onto (ℝ): Since negative values are never obtained as output, the function is not onto when codomain is ℝ.
- Bijective when codomain is [0, ∞): The function becomes onto if we set the codomain to [0, ∞), but it remains non-injective unless the domain is also limited to [0, ∞).

2. Greatest Integer Function: ( f(x) = [x] )

Domain: ℝ
Codomain: ℝ (typically ℤ ⊆ ℝ)
Range: The output values include all integers (ℤ).

Graph: Step-wise horizontal segments. Each segment spans from x ∈ [n, n+1) and maps to the integer n. Discontinuities exist at integer points.

Nature:
- Not one-one: All real numbers in the interval [n, n+1) map to the same integer n.
- Surjective (ℤ): If the codomain is set to ℤ, then for each integer value, there exists a real number x such that the function maps to it, making it onto.
- Not bijective: Not injective, hence not bijective.

3. Signum Function: ( f(x) = text{sgn}(x) )

Definition:

• f(x) = 1, if x > 0
• f(x) = 0, if x = 0
• f(x) = –1, if x < 0

Domain: ℝ
Codomain: ℝ
Range: {–1, 0, 1}

Graph: Three horizontal lines:

• y = –1 for x < 0
• y = 0 at x = 0
• y = 1 for x > 0

Nature:
- Not one-one: All positive values map to 1 and all negative values to –1.
- Onto (if codomain is {–1, 0, 1}): Each value in the codomain has a pre-image.
- Not bijective: Since multiple inputs lead to the same output, not injective.

Conclusion:

The graph of a function is a powerful tool to understand its properties. The modulus function is continuous and symmetric; the greatest integer function is discontinuous and step-wise; the signum function is discrete and piecewise constant. Analyzing graphs helps in identifying function types, ranges, continuity, and whether they satisfy injectivity or surjectivity for a given codomain.