Study on Linear Function Concepts.

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In this lesson let me take you through basically on Linear Function Concepts.

Objective Function: The Objective Function is a linear function of variables which is to be optimized i.e., maximized or minimized. e.g., profit function, cost function etc. The objective function may be expressed as a linear expression.

Constraints: A linear equation represents a straight line. Limited time, labor etc. may be expressed as linear in equations or equations and are called constraints.

Optimization: A decision which is considered the best one, taking into consideration all the circumstances is called an optimal decision. The process of getting the best possible outcome is called optimization.

Solution of a LPP: Linear Function is a set of values of the variables x₁, x₂, �.xn which satisfy all the constraints is called the solution of the LPP..

Feasible Solution: A set of values of the variables x₁, x₂, x₃,�.,x_n which satisfy all the constraints and also the non-negativity conditions is called the feasible solution of the LPP.

Optimal Solution: The feasible solution, which optimizes (i.e., maximizes or minimizes as the case may be) the objective function is called the optimal solution. Important terms Convex Region and Non-convex Sets

Keep reading and may be in the next lesson let me help you on Linear Functions and Slope Forms.

Laws of Exponents

Laws of Exponents

Laws of Exponents are short forms used to represent a number of repeated factors. We use Exponential Notation to represent repeated multiplication. When we say

2 × 2 × 2 × 2 × 2 = 2⁵--- Exponential Notation

2 is the base and 5 is the exponent.

In general for any real number a and positive integer m then a multiplied m times can be written as,

a × a × a × a × a × a ….. m times = a^m

All the Laws of Exponents can be derived from the basic laws of multiplication and division.

Multiplication of Exponents or The Product Rule for Exponents

Let us consider 3³ × 3²

3³ × 3²= 3 • 3 • 3 • 3 • 3 = 3⁵

This can be rewritten as 3³ × 3²= 3³⁺²

= 3⁵

In general, for any real number a and rational numbers, m and n we have

a^m × aⁿ= (a × a × a × a × …..m times) × (a × a × a × a…… n times)

a^m × aⁿ = m factors × n factors

a^m × aⁿ = total of (m+n) factors

a^m × aⁿ = a^m+n

The product of two exponential numbers with the same base is the base raised to the power of the sum of the exponents. Remember, the bases must be same! We cannot use this rule if the bases are different.

Power of Exponents

Consider (3⁴)²

(3⁴)²= (3⁴)× (3⁴)

= ( 3 × 3 × 3 × 3) × (3 × 3 × 3 × 3)

= 3⁸

This can be rewritten as (3⁴)²= 3^4×2= 3⁸

^{Let me show you one example on Law of Exponents:}

Example: Simplify (3⁸ × 3⁴)/3²²

Solution: Using the rule a^m × aⁿ = a^m+n

(3⁸ × 3⁴) = 3⁸⁺⁴

(3⁸ × 3⁴) = 3¹²

Now we have 3¹²/ 3²²

Using the rule a^m ÷ aⁿ = a^m-n

3¹²/ 3²²= 3^12-22 = 3^-10

The solution has a negative power since n > m.

Hence, (3⁸ × 3⁴)/3²² = 3^-10