The following piecewise function gives the tax owed, T(x), by a single taxpayer on a taxable income of x dollars.
T(x) =
(i) Determine whether T is continuous at 6061.
(ii) Determine whether T is continuous at 32,473.
(iii) If T had discontinuities, use one of these discontinuities to describe a situation where it might be advantageous to earn less money in taxable income.
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Tough to describe the function, when it not given...
To determine whether the given piecewise function T(x) is continuous at a specific value, we need to check if the limit of the function exists and is equal to the value of the function at that point.
(i) To determine if T is continuous at x = 6061, we need to evaluate the limit of T(x) as x approaches 6061 from both the left and right sides. We then compare that with the value of the function at x = 6061.
Let's start by finding the limit as x approaches 6061 from the left side. In this case, we evaluate T(x) when x < 6061. Looking at the given piecewise function, T(x) = 0 for x < 6061. So, the limit of T(x) as x approaches 6061 from the left side is 0.
Next, let's find the limit as x approaches 6061 from the right side. In this case, we evaluate T(x) when x > 6061. Looking at the given piecewise function, T(x) = x - 6061 for x > 6061. So, the limit of T(x) as x approaches 6061 from the right side is (6061 - 6061) = 0.
Since the limit of T(x) as x approaches 6061 from both the left and right sides is equal to 0, which is also the value of the function at x = 6061, we can conclude that T(x) is continuous at x = 6061.
(ii) To determine if T is continuous at x = 32,473, we follow the same process.
Evaluating the limit of T(x) as x approaches 32,473 from the left side, we find that T(x) = 0 for x < 32,473. So, the limit of T(x) as x approaches 32,473 from the left side is 0.
Evaluating the limit of T(x) as x approaches 32,473 from the right side, we find that T(x) = 4,989.2 for x > 32,473. So, the limit of T(x) as x approaches 32,473 from the right side is 4,989.2.
Since the limit of T(x) as x approaches 32,473 from the left is 0, and the limit from the right is 4,989.2, which is not equal to each other or the value of the function at x = 32,473, we can conclude that T(x) is not continuous at x = 32,473.
(iii) Since we have determined that T(x) is not continuous at x = 32,473, we can use this discontinuity to describe a situation where it might be advantageous to earn less money in taxable income.
In this case, earning less money in taxable income, specifically below 32,473, would result in a tax owed of 0. However, once the taxable income exceeds 32,473, the tax owed quickly jumps to 4,989.2. Therefore, if someone is in a financial situation where their taxable income is close to 32,473, it might be advantageous for them to earn slightly less to avoid the sudden increase in taxes.
It is important to note that this situation might vary depending on other factors and individual circumstances, and it is always best to consult a tax professional for personalized advice.