Final answer:
The piecewise function for a tax system is not provided in full, and more information is needed to specify income brackets. The given information suggests changes in a consumption function due to taxes, affecting disposable income and consumption.
Step-by-step explanation:
The question appears to revolve around both the calculation of taxes and the concept of a piecewise function. Given that a student's annual income (x) determines their state income tax, the structure of the tax system can be represented as a piecewise function, where the tax amount depends on the income bracket the individual falls into. The options given (a-d) are not a complete piecewise function but seem to represent parts of a larger piecewise function that would specify different tax rates and brackets.
The information provided outside of the options seems to relate to a consumption function and the effect of taxes on disposable income and consumption. A consumption function typically captures the relationship between national income and the amount that households consume. When taxes come into play, they reduce disposable income and therefore can lower consumption per additional dollar earned, affecting the marginal propensity to consume.
To correctly answer the student's question, more context and complete some numerical ranges are needed to define the income brackets for the piecewise function of the state income tax.