In the United States, personal income taxes are paid according to a specific scale. For example, individuals who earn a yearly salary between $31,000 and $31,150 must pay $4229 in taxes. Individuals who earn a yearly salary between $41,000 and $41,150 must pay $6381 in taxes. (Source: irs.gov)

Which type of function would best model this situation: a step function or a piecewise function? Explain

sttep

pisewise function

A step function would best model this situation.

A step function is a function that has a constant value within specific intervals and changes abruptly between those intervals. In this case, the income tax payments change abruptly as the salary falls within different ranges. The tax amount remains constant within each interval but changes abruptly when the income crosses a threshold.

On the other hand, a piecewise function is a function defined by different expressions over different intervals. Although the tax payments vary within different salary ranges, the function itself does not change in terms of its expression. Instead, the function remains the same, with the tax amount depending on the specific range in which the income falls.

Therefore, since the tax amounts for different income ranges are constant and there are abrupt changes between those ranges, a step function would be the most suitable modeling approach in this situation.

To determine whether a step function or a piecewise function would best model the given situation, let's first understand what these functions represent.

A step function is a function that has a constant value within intervals and changes abruptly at the boundaries of those intervals. It "steps" from one value to another.

A piecewise function, on the other hand, is a function that is defined by multiple expressions or equations, each of which applies to a specific interval or range of the input.

In the given situation, we have different tax amounts based on different income ranges. This implies that the tax calculation changes abruptly at specific income thresholds. As a result, a step function would be the most appropriate model.

Here's how you can determine the equation for the step function:

1. Identify the income ranges and their corresponding tax amounts provided in the given statement.

For example:
- Income range $31,000 to $31,150, tax amount $4,229
- Income range $41,000 to $41,150, tax amount $6,381

2. Find the specific thresholds or boundaries within the input domain (in this case, income ranges) where the tax changes.

For example:
- The threshold between $31,000 and $31,150
- The threshold between $41,000 and $41,150

3. Write the step function equation.

For example, let's represent the income as x and the tax payment as f(x).

The step function equation could be written as:

f(x) = (tax amount for the income range) when x is between (lower income limit) and (upper income limit)

For example:
f(x) = $4,229 when x is between $31,000 and $31,150
f(x) = $6,381 when x is between $41,000 and $41,150

Remember to adjust the income ranges and tax amounts based on the data provided in the problem.

By using a step function, you can model the tax situation accurately, as it allows for abrupt changes in the tax amount at specific income thresholds.