Section 1

Functions

Suppose \(y\) and \(x\) are two variables and suppose \(y=\)\(f\)\((x)\). Here, \(f\) presents the rule which assigns to each value of the variable \(x\) to one and only one value of \(y\). To tell it simply, \(y\) is a function of \(x\) if for each value of \(x\) we get one and only one value for \(y\). We generally write the function as \(y=f(x)\) but letters such as \(g\) and \(h\) are also used to present functions.

For example, \(y=10x-5\) is a function because in this example for each value of \(x\) there exists one and only one value of \(y\). Consider another example: \(y=f(x)=x^2+4x-5\). The second example also satisfies the criteria to be a function.

However, there are some relations where \(y\) cannot be considered as a function of \(x\). For example, \(y^2=x+2\) is not a function of \(x\) . This is because \(y^2=x+2\) is equivalent to \(y=\pm{\sqrt{(x+2)}}\). Here, for each value of, \(x>-2\), there exists two values of \(y\) which violates the criteria to be a function.

In a function \(y=f(x)\),
\(x\) is called the input variable, or the independent variable, whereas, \(y\) is the output variable or the dependent variable. In some applications in economics, \(x\) is called the exogeneous variable and \(y\) is the endogenous variable.

Given a function \(f(x)\), the domain is the set of numbers of \(x\) for which the function is defined. For example, if a function is \(y=10-2x\) its domain is the set of all real numbers. On the other hand, the domain of the function \(y=f(x)={{1}\over(x-2)}\) is the set of all real numbers except {\(x=2\)}. This is because if we set \(x=2\), the denominator of the function becomes \(0\) which makes the function undefined. Some functions may have restricted domains. For the function \(y=f(x)={{1}\over(x-2)}\) the domain is restricted based on the mathematical reason that the domain can take any real number except {\(x=2\)}. The reason of the restriction in this example is that a number cannot be divided by zero. Another common mathematical restriction on the domain is that a domain cannot take those values for which the values of the function become imaginary numbers. In economics, certain applications can restrict the domain. For example, if we consider a revenue function \(R(x)\) which is a function of the quantity produced \(x\), the domain would be the set of positive numbers only as the quantity produced cannot be negative.

An inverse function reverses the relationship in a function. If \(y=f(x)\), its inverse function can be written as \(x=g(y)\), if the inverse function exists. For example, if a function is \(y=f(x)=10-2x,\) its inverse function is \(x=g(y)=5-0.5 x\). On the other hand, if a function is \(y=f(x)=x^2\) with an unrestricted domain, its inverse function \(g(y)\) does not exist.