Types of Functions

Introduction:

A function defines a relationship between an independent variable (input) and a dependent variable (output). Different types of functions model different types of relationships in real-world phenomena, such as cost curves, production functions, and growth models.

Types of Functions:

1. Constant Functions

A constant function is a function whose output value remains the same, regardless of the input value. It can be expressed as:

f(x) = c 

Where c is a constant real number.

Example:

f(x)=7 

This means that for any value of x, f(x) will always equal 7.

Graphical Representation:

In the Cartesian coordinate system, a constant function is represented as a horizontal straight line parallel to the x-axis.

2. Polynomial Functions

A polynomial function is an algebraic expression involving a sum of powers of the variable x, each multiplied by a coefficient. The general form is:

Linear Function:

• Form: f(x)=a + bx

• Graph: Straight line with slope b and y-intercept a

Use: Often models relationships with constant rates of change (e.g., cost = fixed cost + variable cost × quantity).

Quadratic Function:

• Form: f(x) = ax² + bx + c, where a≠0

• Graph: A parabola (U-shaped curve)

Use: Models phenomena involving increasing or decreasing returns (e.g., profit maximization, projectile motion, cost minimization).

Cubic Function:

• Form: f(x)=ax³+bx²+cx+d, where a≠0

• Graph: S-shaped curve with possible inflection point

Use: Models complex relationships such as total revenue curves, or demand with changing elasticity.

3. Rational Functions

A rational function is the ratio of two polynomial functions. It has the form:

Where P(x) and Q(x) are polynomials, and Q(x)≠0.

Example:

Special Case:

The rectangular hyperbola is a rational function of the form:

This appears frequently in economics (e.g., demand-supply relationships, isoquants with perfect substitutes).

Graph:

Often has asymptotes—lines the curve approaches but never touches (e.g., vertical and horizontal asymptotes).

4. Exponential Functions

An exponential function has a variable in the exponent. It takes the form:

Where a>0, a≠1, and x is any real number.

Example:

f(x) = 2ˣ

Key Properties:

• If a > 1: function increases rapidly (exponential growth).

• If 0 < a < 1: function decreases rapidly (exponential decay).

• Slope increases as x increases.

Applications:

• Compound interest

• Population growth

        • Technology adoption curves

5. Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It is defined as:

Where:

• b is the base of the logarithm (usually e or 10)

• x > 0

Example:

Special Case:

• Natural logarithm: ln⁡(x)=log_e⁡ x, where e≈2.718

Graph:

• Passes through point (1,0)

• Increases slowly for large values of x

• Undefined for x ≤ 0

Applications:

• Elasticity and utility functions

• Information theory

• Measuring economic scales (e.g., GDP in logs)

• Diminishing returns

Summary Table: