## Abstract

Active learning involves the use of inverse functions where the direct demand function is expressed as an inverse. Functions of two or more independent variables use the coefficient of determination and are expressed as a percentage with a high percentage indicating that the correlation between dependent and independent variables is high. Curvilinear functions consist of mixed elasticity with a downward concave affected by economic effect where alternative commodities make demand function to be more elastic. Linear function occurs due to parallel shift in demand which allows demand curves to be illustrated using linear demand functions and linear supply curves have parallel shifts in supply.

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## Inverse functions

This is expressing the price of a commodity as a function of the quantity demanded because mathematically, this is a direct function of demand expressed as an inverse. When Y is expressed as a function of X, the inverse function is X expressed as a function of Y where both X and Y are direct and inverse functions. The direct function becomes inverse if it is expressed in the opposite direction and can still return back to its initial set. A function is invertible if it has an inverse relationship that is a function where inverse relation forms inverse function. The inverse function identifies input X in a unique way basing it on output and the invertible function has a rule that defines function. For example Q*d* =700 -10p. When the direct equation for demand is solved, p becomes the inverse demand.

The Law of demand is valid if goods and services offered for sale tend to be bought in large quantities when their prices are high holding all other things constant. For example, a customer goes to the market hoping to purchase goods for $20 but notices the price has risen to $30 and goes ahead to purchase an extra quantity. Generally, sellers do not advertise an increase in the price of goods and services for the sales to increase or dispose of unwanted inventory.

Direct demand function where the quantity demanded is a function of price can be gotten from demand function in general where customers change the quantity of a commodity demanded due to price change because the higher the price of the commodity the low the demand and the low the price of the commodity the higher the demand. (Webster, 2005 pp33-37)

## Functions of two or more independent variables

When given a dependent variable, a fraction of total variation is measured using the coefficient of determination where any change in sales is explained by changes in expenses for advertisement. If it is costly to make advertisements in order to create awareness to customers and the advertisements make more of the commodity to be purchased, then sales will increase. The coefficient of determination is expressed as a percentage since if it is 0.65; it means 65% of the variation. When the percentage is high, a correlation that exists between dependent and independent variables becomes high.

The Scatter diagram has observations lying close to the regression line because of small deviations existing from the line and high correlation between Y and X. In cases of extreme situations, observations are on a straight line. When two variables have a high correlation between them or regression coefficient, the variation independent variable does not come as a result of variation in the independent variable because, it may happen due to another cause such us change in Z where X is correlated to Z making correlation to exist between Y and X, then, change in X can not change Y.

The coefficient of determination is used to measure the fraction of overall variation in Y is explained by variation in X and its value ranges from 0 to 1. A high coefficient shows that there is a high correlation between X and Y and the scatter diagram fits the regression line. F-statistics test the importance of regression equation and give a ratio of variation independent variable to variation that is not explained. The importance of the equation is determined by comparing the statistics obtained with critical values using significance level and degree of freedom. The main aim is to have knowledge of setting up regression and interpreting the findings. (Julian, 2006 pp23-26)

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## Curvilinear functions

Curvilinear functions have mixed elasticity and concave downward with elastic and inelastic demand. Economic effect affects the shape of curvilinear demand function and demand function becomes more elastic with the availability of alternative commodities than the ones in closed economic conditions. The formula for computing the elasticity of linear demand can be used to calculate the point of elasticity in curvilinear demand. (Bayle, 2004 pp20-25)

## Linear functions

Linear demand functions are caused by demand having parallel shifts which make demand curves to be illustrated using linear demand functions. Linear supply curves bring about supply having parallel shifts. Supply and demand curves are perfectly linear and their shifts are parallel providing a way of supply analysis and learning demand with linear functions being used to approximate the curves. When there shift in demand or supply and the lines are not parallel, the new curves do not cut across original supply and demand curves.

Linear functions have their input variable X raised to first power and yields to graphs with straight lines. Linear functions have slope or gradient and Y-intercept is the intersection point between graph and Y-axis, in case of change in real constants, the line may become shallower or steeper and change in Y-intercept makes the line move downward or upward. (Krugman, 2005 pp14-16)

## Relating the description of each function to appropriate economic concept

### Inverse function

Real growth in the economy is studied quantitatively through the use of national income and aggregate inputs which is responsible for growth in the long run and economic efficiency. Deviation in economic growth has a mean value of zero in studying forces of business cycle while demonstrating the understanding of real growth processes and explaining the behavior of two components. Exogenous shocks are the forces that drive fluctuations of growth rate in gross domestic product. Real per capita growth has increment in growth at a constant rate and fluctuations are shown by the change in population component.

The gross domestic product per capita grows with time in a straight line if a change in a given age population is small. The relative rate of growth in the gross domestic product is an inverse function of the real gross domestic product with a constant numerator of economies that have already developed.

People are considered as economic agents who produce money and their distribution is according to a hierarchical structure that has a rigid response to external disturbances such as economic growth and inflation. The relative income is in predefined distribution where a small portion of the entire population gets a portion of nominal or real income and all the positions in distribution have individual agents who occupy them and propagate to different income positions with vacant places getting someone to fill. This brings about change in equilibrium with the stable distribution of personal income.

Every person by definition can produce money the same way he gets income. This helps to balance earnings and production bringing about local balance. Gross domestic product is the personal income of everybody where it formulates income side and equilibrium as a measure of goods and services produced in order to define real income produced by a given population in a specified age structure. In every society, individual income is indicated by the relationship between gross domestic product per capita and age to predefine growth in potential income.

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The difference between observed and the predicted growth is induced using stochastic exogenous shock and the difference is a result of effects that can not be controlled due to unknown forces. There are two components of real economic growth where the model is verified by econometrical and statistical tools validated by an additional set of data. (Krugman, 2005 pp17-19)

### Functions of two or more independent variables

There is a relationship between short-run employment in manufacturing firms and non-manufacturing firms. This helps to examine how changes in employment of non-manufacturing firms can associate with employment changes in manufacturing firms. There are lags where two sectors of employment are relating which have a quantitative significance due to the assumption that the manufacturing sector is able to exports most of its manufactured products and demand may fluctuate as a result of external factors. Non-manufacturing sector experiences fluctuation in employment and demand due to changes in the manufacturing sector.

Correlation analysis examines association in how the two sectors vary to enable it to detect leads and lags. The analysis suggests that variation between the two variables has strong simultaneity reflecting underemployment and unemployment in industries thereby influencing fluctuations. Distributed lag function investigates quantitative significance in any relationship where it suggests that when the number of workers in the manufacturing sector change, there will also be a change in the number of workers of the non-manufacturing sector but there is variation in the effect. (Bhaskar, 2004 pp45-47)

### Curvilinear functions

Performance of a firm can be presented as curvilinear in both objective and perpetual performance where the study shows a relationship inverted in a U-shape. High entrepreneurial orientation firms are innovative and engage in risky ventures by being proactive in pursuing the available opportunities. For example, a firm must be a risk-taker in order to get a high return from its investment because there is a positive relationship between the performance of a firm and the relationship with the entrepreneur. The strategic posture of entrepreneurship leads to inconsistent improvement in performance due to neglect of the market context where businesses are operated.

Internal structures of organizations and their institutional environment have characteristics different from mature economies. Experienced management teams benefit organizations by transforming them from a planned economy into a market economy. Market-based systems and socialists coexist with the government taking control of the resources and distributing materials which promote the relationship between the performance of the firms and entrepreneurial orientation.

A curvilinear relationship explains a positive relationship where the firms put effort to be innovative and take risks to pursue opportunities in the market and enhance the performance of the firm. However, some firms are not equal in their operations and their relationship is expressed as a curvilinear function because they do not have formalized organization and institutional support required for a firm to undertake innovative and risky projects.

Entrepreneurial orientation is affected by the capability of the firm, strategic processes, and availability of capital which is different for market economies that are affected by private businesses. At saturation point, firms can not perform well due to entrepreneurial orientation, and firms with high entrepreneurial orientation are not linear but curvilinear. The institutional theory explains the behavior of organizations and decision-making by management teams in the economies which are emerging. Emphasis is on how the system’s strategic posture helps to shape the organization through control and market coordination. In cases where contradictions exist, it results in confusion to the entrepreneurial firms making them operate under stress without engaging in new businesses efficiently. (Blaug, 2003 pp12-13)

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### Linear functions

The important economic concept is the way economic agents make their decisions and the importance of the decisions to themselves and other people. Linear functions focus more on the relationship between tax policies of the government and revenues to be generated through the efforts put into work. The government makes choices because it may be dangerous if the relationship between taxation and revenue collected is non-linear.

The government makes pledges during elections which results to demand more money to meet the pledges such as paying fees for higher education. When elections are over, it is upon the government to offer all the pledges which require large sums of money. This money can only be generated through taxation where the people who earn more income pay more taxes and people with low-income pay less tax which is expressed as a percentage of their total earnings.

The government must be able to balance its income and expenditure so that the income is able to take care of all the expenditures in the budget. If the income generated is less than the expenditure the government encounters a budget deficit and may result in borrowing in order to finance the deficit. A linear function is related to economic concepts because the government uses its fiscal policies of taxation and spending where it generates tax to finance its expenditure every financial year by making sure that it is able to balance its income with the expenditure to be incurred. (Blaug, 2003 pp14-16)

## Interpretation and analysis of why and how each function helps us better understand economics

### Inverse functions

(Freund, 2004 pp25-26) found that, Dynamic process of equilibrium, in the long run, is demonstrated in a market setting where the seller decides whether to supply a commodity in the market by depending on the cost of production and selling price of the product. Each product has its own cost of production and the price to be charged for each product depends on the amount of supply and quantity demanded by customers. Inverse demand curves help to determine the number of customers that are expected to purchase the product at any given time at a specified market price.

The sellers in the market strive to maximize profit through ensuring that there is free entry into the market and free exit where the profits will be equal across markets and opportunity cost will help to determine the profit to be generated by revealing how to attain equilibrium in the long-run. The seller is able to know the best market to enter in a situation where opportunity cost is zero and each market has zero accounting profits. The inverse function helps to understand how a competitive market is able to clear and make competitive equilibrium efficient. Economic profits are defined using short-term and long-term profitability that help the firm to compete with similar producing firms by improving the quality of products and charging affordable prices to potential customers.

Long-run equilibrium assumes equal profits because every producer seeks to make a profit by ensuring that marginal revenue is equal to marginal costs and no losses are made at the end of the production period because it requires only a few sellers who are profit-seekers for the profits to be equal across markets. (Freund, 2004 pp27-28)

### Functions of two or more independent variables

Variance explains how function can predict dependent variables without using the mean value. When the equation of the data is not fitted and all information is ignored, the prediction for the value of the dependent variable in the observation is a mean value obtained in all the observations.

The variance is obtained by squaring the differences and looking for their sum. If the fitted function is used to predict value instead of using the mean value of a dependent variable, the second variance is calculated using a square of the predicted variable and actual value. In situations where the function predicts the observed data perfectly, the value of statistics will be 100% and if it does not predict the dependent variable except by use of mean, the value is 0.00.

In economics, regression applications use data in time series where the straight line has periodic data with periodic deviations which are positively correlated. Autocorrelated deviations have the function fitted with inappropriate values of data such as linear equations in quadratic data. When a standard error in the estimated value of parameter underestimates the standard error, intervals of confidence will be incorrect. Statistics help to test the importance of the regression model by testing a null hypothesis with zero regression coefficients. (Zevin, 2004 pp14-18)

### Curvilinear functions

Curvilinear functions help in knowing how to use graphs in establishing a linear relationship between variables through examining relationships along with predictor variables. For example, in looking at spouses’ personal income, we consider men between thirty to sixty-five years in order to omit teenagers and retired men and men with personal income not more than zero. This will help to show that as age goes up, personal income increases but declines at a later age. During the early stages of life, the spouses are energetic and able to work hard to earn extra income but later in life, there is reduced activity due to aging and less income is earned. (Weaver, 2007 pp25-29)

### Linear functions

Linear functions help to understand differential equation theory in economic analysis and deals with dynamic systems of economic analysis. Linearity determines how the operation of a system tomorrow can not be determined by the way it is operating today. The linear function helps to simplify a problem in order to find a solid solution. Nonlinear dynamics studies evolution equations that are nonlinear and in case of change in parameter for a linear system, there will be no change in quality or nature of behavior.

Changes in the parameter of a nonlinear system result in dramatic or sudden changes in qualitative and quantitative behavior in a system. Dynamical theory shows the way interactions lead to qualitative structures and their relationship with individual components. The complexity of ideas is applied in scientific disciplines such as economics where common properties are shared to unify the methodological factors for exploring structures in different systems. (Kappa, 2005 pp34-37)

## References

Webster T. (2005): Theory and practice in managerial economics: Oxford university Press pp. 33-77.

Julian T. (2006): Application of managerial economics: Harvard University Press, pp. 23-26.

Bayle M. (2004): Business strategy and managerial economics: McGraw-Hill, pp. 20-25.

Krugman P. (2005): Theory and policy on international economics: Addison Wesley, pp. 14-19.

Bhaskar N. (2004): Economic concepts in Micro-economic theory: Routledge, pp. 45-47.

Blaug M. (2003): Who is who in economics: Edward Elgar, pp. 12-16.

Freund J. (2004): The sociology of Max Weber: Pantheon Books, pp. 25-28.

Kappa D. (2005): Economic concepts and approaches: Hollywood, pp. 34-37.

Weaver F. (2007): Economic literacy; Basic economics: John Wiley and sons, pp. 25-29.

Zevin J. (2004): Managerial review of economic concepts: American management association, pp. 14-18.