Algebraic Functions

Types and Behaviors

1. Identity Function

Form: $f(x) = x$

Behavior:

• Passes through the origin $(0, 0)$;
• A straight line with slope $m = 1$;
• Every input maps to itself;
• No maximum or minimum;
• Domain: $(-\infty, \infty)$;
• Range: $(-\infty, \infty)$.

Use case: representing unchanged data or as a reference in transformations.

2. Constant Function

Form: $f(x) = c$

Behavior:

• A horizontal line at $y = c$;
• The output remains constant for all inputs;
• Slope: $m = 0$;
• No maximum or minimum;
• Domain: $(-\infty, \infty)$;
• Range: ${c}$.

Use case: representing fixed quantities such as baseline values or flat fees.

3. Linear Function

Form: $f(x) = mx + b$

Behavior:

• A straight line with slope $m$;
• Increasing if $m > 0$, decreasing if $m < 0$;
• X-intercept: $x = -\frac{b}{m}$;
• Y-intercept: $y = b$;
• No maximum or minimum;
• Domain: $(-\infty, \infty)$;
• Range: $(-\infty, \infty)$.

Use case: predicting continuous outcomes such as revenue or costs.

4. Polynomial Function (Quadratic Example)

Form: $f(x) = ax^2 + bx + c$

Behavior:

• Parabolic curve (U-shaped if $a > 0$; inverted U if $a < 0$);
• Vertex at $x = -\frac{b}{2a}$;
• X-intercepts (roots): $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$;
• Y-intercept: $f(0) = c$;
• Domain: $(-\infty, \infty)$;
• Range:
  • If $a > 0$, then $[y_{vertex}; \infty)$;
  • If $a < 0$, then $(-\infty; y_{vertex}]$.

Use case: curve fitting, regression models, and describing non-linear trends.

5. Rational Function

Form: $f(x) = \frac{p(x)}{q(x)}$

Example: $f(x) = \frac{1}{x - 1}$

Behavior:

• Vertical asymptote at $x = 1$;
• Horizontal asymptote at $y = 0$;
• Undefined at $x = 1$;
• Sharp increase and decrease near the asymptote;
• Domain: $(-\infty, 1) \cup (1, \infty)$;
• Range: $(-\infty, 0) \cup (0, \infty)$.

Use case: modeling constrained systems such as rates of change or resource utilization.