Question

A union contract guarantees a \(9 \%\) yearly increase for 5 years. For a current salary of $$\$ 28,500$$, the salaries for the next 5 years are given by \(S=28,500(1.09)^{[t]}\) where \(t=0\) represents the present year. (a) Use the greatest integer function of a graphing utility to graph the salary function, and discuss its continuity. (b) Find the salary during the fifth year (when \(t=5\) ).

Short answer

The graph of the salary function is a sequence of separating horizontal lines due to the annual increase in the salary, leading to discontinuities at each start of the year. The salary in the fifth year is obtained by computing the given expression \(S = 28,500(1.09)^{5}\).

Step-by-step solution

Plotting the Function

To graph the given function \(S=28,500(1.09)^{[t]}\), plot the graph on a graphing utility. Variable \(t\) represents time in years on the x-axis and \(S\) is the salary in dollars on the y-axis. The plot would be a sequence of five straight horizontal lines each representing the salary for each year, because the salaries are increased annually, hence they are constant during each year. Since the function takes integer values for time, and salary jumps at the start of each year but maintains its value throughout the year, this function is discontinuous at each start of the year.

Calculate the Salary in the Fifth Year

To determine the salary during the fifth year, substitute \(t=5\) into the function. \(S = 28,500(1.09)^{5}\). Calculate this expression to get the salary during the fifth year.

Key concepts

Greatest Integer Function

The greatest integer function, also known as the floor function, is a mathematical function that maps a real number to the largest integer less than or equal to it. It is denoted by \(\lfloor x \rfloor\) or, in some contexts, as \[t\], where \(t\) represents a real number.

When you graph a greatest integer function, you'll see a series of steps or plateaus because the output value is held constant until the next integer is reached. This is why it is also sometimes called a 'step function.' The function can be represented visually as a series of horizontal line segments, and when graphed using a graphing utility, it offers a powerful visual representation of situations where values are incremented in whole numbers.

In the context of the provided exercise, the greatest integer function is used in the salary formula to represent the yearly salary at the start of each year, rounded down to the nearest whole year. This perfectly models the scenario where salary is calculated annually as opposed to continuously.

Graphing Utility

A graphing utility is a tool—often a software program or a feature within a calculator—that allows users to plot mathematical functions and analyze their behaviors visually. It is particularly helpful for understanding the nature of different kinds of functions, like linear, polynomial, or in our case, greatest integer functions.

Graphing utilities can manage complex calculations and plot them accurately, which is why they're used extensively in educational settings to help students visualize mathematical concepts. To graph the salary function in our exercise, the graphing utility considers the greatest integer function, representing time in whole years, and allows for the visualization of the discontinuous salary increments over the span of the employment contract.

Discontinuous Function

A discontinuous function is a function that is not continuous at one or more points in its domain. In simpler terms, there are breaks or jumps in the graph of the function where it does not flow smoothly from one point to another.

The salary function in our exercise, \(S=28,500(1.09)^{[t]}\), is an example of a discontinuous function. As the salary only increases at the beginning of each year, the graph representing the salary over time includes jumps at the end of each year. Graphically, these jumps would appear as sudden increases on the graph at each new year, reflecting the salary increment, followed by a flat line as the salary remains constant through the year.

Exponential Growth

Exponential growth is a process that increases quantity over time at a rate proportional to its current value. This pattern results in a rapid increase of the quantity as time progresses. In mathematical terms, exponential growth is described by the function \(f(t) = a(1 + r)^t\), where \(a\) is the initial amount, \(r\) is the rate of growth, and \(t\) is time.

In our exercise, the salary grows exponentially at a rate of 9% per year. However, due to the greatest integer function, this growth only takes effect at the start of each year when the salary is recalculated. Therefore, the growth isn't continuous over time but rather happens in discrete jumps. Despite this, over the long term, as seen during the five-year period, the overall trend of the salary is an exponential increase.