Question

The hourly wage increase each employee receives each year depends on their number of years of service. Every three years of service means an increase of $0.50 per hour. So, employees that have been with the company for less than three years can expect to receive an increase of $0.50 per hour. Employees that have been with the company for at least three years, but less than six years can expect an increase of $1.00. Employees that have been with the company for at six years, but less than nine years, receive an increase of $1.50 per hour. And, employees of at least nine years, but less than twelve years receive an increase of $2.00. Write a function to represent this scenario.

Which graph represents this wage increase for x < 12?

Answer 1

Answer: The Function is,

`f(x) = \begin{cases}0.50 & \text{ if } x<3 \\ 1.00 & \text{ if } 3\leq x<6 \\1.50 & \text{ if } 6\leq x<9 \\ 2 & \text{ if } 9\leq x<12\end{cases}`

The below graph the wage increase for x < 12

Explanation:

Let x represents the number of years and f(x) represents the increment in wages.

According to question,

For the employees that have been with the company for less than three years can expect to receive an increase of $0.50 per hour.

That is, for x < 3, f(x) = 0.50

Employees that have been with the company for at least three years, but less than six years can expect an increase of $1.00.

That is, for 3 ≤ x < 6 , f(x) = 1.00

Employees that have been with the company for at six years, but less than nine years, receive an increase of $1.50 per hour

That is, For 6 ≤ x < 9, f(x) = 1.50

And, employees of at least nine years, but less than twelve years receive an increase of $2.00.

That is, For 9 ≤ x< 12, f(x) = 2.00

Thus, the complete function that shows the total increment in wages is,

`f(x) = \begin{cases}0.50 & \text{ if } x<3 \\ 1.00 & \text{ if } 3\leq x<6 \\1.50 & \text{ if } 6\leq x<9 \\ 2 & \text{ if } 9\leq x<12\end{cases}`

Which is the piece wise function.

Since, f(x) = constant for 4 intervals.

Thus, In the graph we will obtain the lines parallel to x-axis.

Answer 2

If we let x as the number of years of service in the company and f(x) as the increase in the wage, the step wise function that describes the scenario is
f(x) = { 0.5, x < 3
{ 1.0, 3 ≤ x < 6
{ 1.5, 6 ≤ x < 9
{ 2.0, 9 ≤ x < 12

The point (2, 12) represents the wage increase of x < 12