Revenue Function: R(x)=xp
One common relation between p and x is an equation of the form;
px=a, where a is a positive constant.
1.) With the horizontal axis representing x and the vertical axis representing p, sketch the general shape of the demand curve associated with this relationship.
2.) What is the smallest feasible value of x in this relationship?
3.) What value of p corresponds to the smallest feasible value of x?
4.) Is there a smallest feasible value for p in this relationship?
5.) Find a formula for the revenue function R(x).
6.) What class of function is R(x)?
7.) How would you describe the graph of R(x)?
consider the graph xy=1
This is just a hyperbola in the 1st quadrant, with horizontal and vertical asymptotes.
1.) The demand curve in this relationship is a rectangular hyperbola. It starts at a positive value on both axes and approaches zero as x or p approaches infinity.
2.) The smallest feasible value of x in this relationship is 0. Since a cannot be negative and p is positive, the only option is for x to be zero.
3.) The value of p corresponding to the smallest feasible value of x (which is 0) is a. Since px = a, when x = 0, p must equal a to satisfy the equation.
4.) There is no smallest feasible value for p in this relationship. As long as a is positive, any positive value for p can be chosen, and x will adjust accordingly to satisfy the equation px = a.
5.) The revenue function R(x) is given by R(x) = xp.
6.) R(x) is a linear function. This is because the function is of the form R(x) = mx + b, where m represents the constant p and b represents the constant 0.
7.) The graph of R(x) is a straight line that passes through the origin in the Cartesian coordinate system. The slope of the line is given by the constant p, and the line extends indefinitely in both the positive x and positive R(x) directions.
1.) To sketch the general shape of the demand curve associated with the relationship px=a, we can rearrange the equation to solve for p: p = a/x. Since a is a positive constant, p will always be positive. The graph of p = a/x will be a rectangular hyperbola, with the x-axis as an asymptote. As x approaches infinity, p approaches 0, and as x approaches 0, p approaches infinity.
2.) The smallest feasible value of x in this relationship is when x is equal to zero. However, this value of x would make p undefined, as division by zero is undefined. Therefore, the smallest feasible positive value of x is any value greater than zero.
3.) Since the smallest feasible value of x is greater than zero, when x is at its smallest feasible value, p will be equal to a divided by that smallest feasible value of x. So, p = a/(smallest feasible value of x).
4.) No, there is no smallest feasible value for p in this relationship. As x approaches zero, p approaches infinity. Therefore, there is no lower limit for p.
5.) The revenue function R(x) is given as R(x) = xp.
6.) The revenue function R(x) is a linear function. It is a function of the form y = mx, where m is the constant coefficient of x.
7.) The graph of the revenue function R(x) will be a straight line passing through the origin. As x increases, the revenue increases proportionally. The slope of the line represents the constant rate of increase in revenue per unit increase in x.