Maths

Introduction to Relationships and Functions

Definition of a Relationship

In mathematics, we begin with a relationship between two variables, namely Y and X. A relationship does not initially qualify as a function until it meets specific criteria that define functions.

Domain and Range

  • Domain: Refers to the set of x-values which function is defined.

  • Range: Refers to the set of y-values corresponding to the x-values in the domain.

A clear understanding of domain and range is essential. For example, if you cannot accurately calculate or determine domain or range in an exponential function, it raises significant questions about your understanding of the foundational concepts.

Demand Function

Explanation of Demand Function

The demand function illustrates the relationship between price (P) and quantity (Q) demanded. For instance, if we have a demand function expressed as:
P=f(Q)P = f(Q)
Where, P equals the price, and Q equals the quantity.

  • If given that Q equals 20, according to the demand function, we can deduce:
    P=100imesrac1QP = 100 imes rac{1}{Q}
    Thus, substituting gives:
    P=100imesrac120=5P = 100 imes rac{1}{20} = 5
    This implies that when the quantity is 20, the price paid is 5 units.

Supply Schedule

Supply Function Overview

In the context of supply, we examine the relationship between price per unit and quantity supplied. For example, when the price per unit is:

  • $500 then the quantity supplied is 11 units.

  • $600 corresponds to 14 units.

  • $700 corresponds to 17 units.

  • $800 corresponds to 20 units.

Function Representation of Supply

This relationship can be illustrated as function evaluations:

  • F(500)=11F(500) = 11

  • F(600)=14F(600) = 14

  • F(700)=17F(700) = 17

  • F(800)=20F(800) = 20

Where F represents the supply function as a function of price. Conversely, we can solve for price (P) given any quantity supplied (Q).

Equal Functions

Definition of Equal Functions

Functions F and G are said to be equal when:

  • The domains of F and G correspond exactly, denoted as:
    F=GextifD(F)=D(G)F = G ext{ if } D(F) = D(G)
    Where D(D) indicates the domain set. This indicates that for every x in the respective domains, F(x) must equal G(x).

Example of Equal Functions

For example, consider the following functions:

  • F(x)=x+2x−1F(x) = \frac{x+2}{x-1}

  • G(x)=x+2G(x) = x + 2

  • H(x)=x+2extwherex<br>eq1H(x) = x + 2 ext{ where } x <br>eq 1

  • K(x)=x+2extforx<br>eq1K(x) = x + 2 ext{ for } x <br>eq 1

Analysis of Domains

To explore equality:

  • The domain of F excludes x = 1, as this would cause a division by zero. The set of all other real numbers constitutes the domain of F.

  • G and H, having no restrictions on their evaluation, have the domain of all real numbers.

Conditions on Function Behavior

Identifying Restrictions and Domains

  • For function H:
    H(x)=1x−1H(x) = \frac{1}{x-1}

  • The condition for domain relevance ensures values do not lead to undefined operations. It must hold that:
    x - 1 > 0 ext{ which implies } x > 1

  • Therefore, the domain of each function must be accurately identified as needing concrete definitions.

Comprehensive Exam of Various Functions

Rational Functions

Rational functions dictate certain behavior, especially with roots and denominators that rely on specific input values. Key principles are:

  • Discriminants must be positive where applicable.

Conclusive Notes on Functions

There are practical implications when examining functions, such as determining their domains, limits, and behaviors as well as establishing values not viable for calculations. Adequate insight into functional relationships and their graphical representations is crucial for mastering these concepts.