Problem 56 - College Algebra

Question

During the 1980 s, 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 $$ f(x)-\frac{80 x-8000}{x-110}, 30 \leq x \leq 100 $$ 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. CAN'T COPY THE GRAPH a. Find and interpret \(f(30)\). Identify the solution as a point on the graph of the function. b. Rewrite the function by using long division to perform $$ (80 x-8000) \div(x-110) $$ Then use this new form of the function to find \(f(30) .\) Do you obtain the same answer as you did in part (a)? c. Is \(f\) a polynomial function? Explain your answer.

Step-by-Step Solution

Answer

a. When the tax rate is \(30%\), the government's tax revenue is approximately $54.5 billion. The point on the graph is \(30, 54.545\). b. The rewritten function after long division is \(f(x) = -80 + \frac{7200}{x - 110}\). The value of \(f(30)\) is the same in both forms of the function. c. \(f(x)\) is not a polynomial function because it contains a negative power of \(x\).

Step 1: Understanding the function

The function given, \(f(x) = \frac{80x -8000}{x - 110}\) represents the government tax revenue, in tens of billions of dollars as a function of the tax rate, \(x\).

Step 2: Calculate and interpret \(f(30)\)

To find \(f(30)\), substitute \(x = 30\) into the function: \(f(30) = \frac{80*30 - 8000}{30 - 110} = 54.545\). Thus, when the tax rate is \(30%\), the government's tax revenue would be approximately $54.5 billion. The solution can be identified as a point on the graph where \(x = 30\) and \(y = 54.545\).

Step 3: Rewrite the function

Use long division to rewrite the function \(f(x) = \frac{80x - 8000}{x - 110}\). After the long division of \(80x - 8000\) by \(x -110\), the new form of the function is \(f(x) = -80 + \frac{7200}{x - 110}\).

Step 4: Verification

Substitute \(x = 30\) into the new form of the function \(f(x) = -80 + \frac{7200}{30 - 110}\) to find \(f(30)\). The obtained value should be same as the value obtained in Step 2. \(f(30) = -80 + \frac{7200}{30 - 110} = 54.545\).

Step 5: Polynomial function check

A polynomial function is an expression that involves only non-negative integer powers of \(x\). The given function \(f(x) = \frac{80x -8000}{x - 110}\) is not a polynomial function because it contains a negative power of \(x\) (it's in the denominator).