Inverse Equations Problem \"A\" Is an Example

Inverse Equations

Problem "a" is an example of substitution equations. In this case, it is asking the two equations to be subtracted and the number four substituted for "x." In order for this operation to be properly accomplished, order of operations requires that the two problems be simplified to their final state before being combined.

Proof a.

(f-h)(4)

Problems "b" are examples of inverse functions. Here, a secondary function is being placed in the original function's "x" position. Once this is accomplished, order of operations and rules of simplification allow for the simplifying of the final expression. These cannot be entirely solved as there is no value given for "x."

Proof b.

2(x^2 -- 3) +

2x^2 -- 6 +

2x^2 -- 1

(7 -- (x^2 -- 3)) /

(7 -- x^2 + 3) /

(10 -- x^2) /

Problems "c" are examples of finding the inverse of a single function. According to inverse theory, this can be accomplished by substituting "y" for "f (x)" and then solving for x. All common rules of algebra must be applied during the solving resulting in the inverse equation.

Proof c.

f (x) = 2x + 5

y = 2x + 5

y -- 5 = 2x

(y -- 5) / 2 = x f (x)^-1= (x -- 5)/2

h (x) = (7 -- x)/3

y= (7 -- x)/3

3y= 7 -- x

3y -- 7 = -x h (x)^-1 = -3x + 7

As with anything in mathematics, linear expressions are essential in the "real" world. The most common example can be found in business where there are multiple fluid factors and a correct proportion must be decided. For example, last week's word problem presented an ideal business scenario where a company is attempting to maximize profit by ensuring the correct amount of employees are present at any given time. This same technique is used in a larger scale by call centers. In the call center setting, it is essential for the efficiency of the center to be maintained at all times of day and call volumes, thus call centers will create formulas based on these simple mathematical principles and apply them to computer program allowing constant computation of workplace efficiency.