In your first job after graduating from college you make ...

In your first job after graduating from college you make $\$ 36,000$ a year before taxes. After four years you get a raise of $\$ 2,500$. Two years after that you change jobs and go to work for a company that pays you $\$ 49,000$ a year. (a) Construct a piecewise-defined function that describes your pretax income in the year that is $t$ years after you graduate from college. (b) Write down a function that describes the total amount of money you will have earned $t$ years after graduating from college. (c) How many years after graduating from college will you have earned a total of one million pre-tax dollars?

In your first job after graduating from college you make $\$ 36,000$ a year before taxes. After four years you get a raise of $\$ 2,500$. Two years after that you change jobs and go to work for a company that pays you $\$ 49,000$ a year.
(a) Construct a piecewise-defined function that describes your pretax income in the year that is $t$ years after you graduate from college.
(b) Write down a function that describes the total amount of money you will have earned $t$ years after graduating from college.
(c) How many years after graduating from college will you have earned a total of one million pre-tax dollars?

Key Concepts

Piecewise-Defined Functions

A piecewise-defined function is one that is described by different expressions over different parts of its domain. This concept is crucial when modeling situations where the behavior of a function changes at certain points, such as changes in job salary over time. The model is built by partitioning the time variable into intervals, with each interval having its own rule or formula.

Function Modeling

Function modeling involves creating mathematical expressions to represent real-life situations. In this context, it means representing a salary progression as a function of time, incorporating events like raises and job changes that alter income. It requires accurate mapping of when and how the changes occur in order to reflect realistic earnings over time.

Linear Functions

A linear function is a function that describes a constant rate of change. In the context of salary functions, incomes over specific time intervals are often modeled as constant or increasing at a constant rate, which can be represented using linear equations. This helps in understanding and computing changes such as fixed raises or promotions.

Cumulative Sums and Integration

Calculating the total amount of money earned over time involves summing up earnings, which is conceptually similar to computing the definite integral (or the cumulative sum) of a function. This involves adding up the contributions from each time interval defined by the piecewise function, keeping track of the individual segments where the rate of income might differ.

Solving Equations in a Piecewise Context

Solving equations in a piecewise context involves determining the value of the variable that satisfies the equation across different segments of the function. This is especially important when finding a particular cumulative total (such as one million dollars) because the equation might take different forms depending on which segment of the piecewise function the total falls into.

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