A general form demand function, \(Q_d=f(P)\), tells that the quantity demanded of a good \((Q_d)\) depends on its market price \((P)\).
A specific form for a demand function, for example, \(Q_d=100-2P\), tells us exactly how the values of the dependent variable, \(Q_d\), are determined from the values of the independent variable, \(P\) . In this example, the specific functional form tells that a 1-unit increase in the price of the good, \(P\), will lead to a decrease in the quantity demanded of the good, \(Q_d\), by exactly \(2\) units.

When \(Q=10\),   \(TC=100+2(10)=120\)

When \(Q=20\),   \(TC=100+2(20)=140\)

Fig. 1(a) presents a function. Fig. 1(b) does not display the graph of a function as for some \(x\) in the domain, we get two values for \(y\).

The domain of the function \(y=\sqrt{(x-10)}\) is all \(x≥10\) as for any \(x<10\) the function takes the square root of a negative number which is an imaginary number.

As production cannot be negative, domain is the set of all non-negative numbers.

$$y = f(x) = 10 + 2x$$ $$y - 10 = 2x$$ $$x = 0.5y - 5$$ The inverse function is: \(x = g(y) = 0.5y - 5\)

The Inverse function does not exist. When we consider, \(y=f(x)=x^2\), for each value of \(x\), we get one and only one value for \(y\). Therefore, \(y=x^2\) is eligible to be a function. However, when we try to find its inverse function, we get \(x=\pm \sqrt{y}\). That means for each positive value of \(y\), we get two values for \(x\). Therefore, in this case, the inverse function does not exist.

$$Q = 10 - 2P$$ $$2P + Q = 10$$ $$2P = 10 - Q$$ $$P = 5 - 0.5Q$$