Question

During the 1980s, the controversial economist Arthur Laffer promoted the idea that tax increases lead to a reduction in government revenue. Called supply-side economics, the theory uses functions such as

f(x) = (80x-8000)/(x-110) , 30 ≤ x ≤ 100
This function models the government tax revenue f(x) in tens of billions of dollars, in terms of the tax rate x. The graph of the function is shown. It illustrates tax revenue decreasing quite dramatically as the tax rate increases. At a tax rate of (gasp) 100%, the government takes all our money and no one has an incentive to work. With no income earned, zero dollars in tax revenue is generated.
Use function f and its graph to solve Exercises 55–56.
a. Find and interpret f(30). Identify the solution as a point on the graph of the function.

Answer 1

To find f(30), we substitute x = 30 into the function f(x) = (80x - 8000)/(x - 110):

f(30) = (80(30) - 8000)/(30 - 110)

Simplifying further:

f(30) = (2400 - 8000)/(-80)

f(30) = -5600/(-80)

f(30) = 70

Therefore, f(30) = 70.

Interpretation:

The value of f(30) represents the government tax revenue in tens of billions of dollars when the tax rate is 30%. In this case, the tax revenue is 70 billion dollars.

This solution (30, 70) represents a point on the graph of the function f(x), indicating that when the tax rate is 30%, the government tax revenue is 70 billion dollars

Explanation: