Question

Give the domain and range of the functions described. Let \(P=v(t)\) give the total amount earned (in dollars) in a week by an employee at a store as a a function of the number of hours worked. The employee earns \(\$ 7.25\) per hour and must work 4 to 6 days per week, from 7 to 9 hours per day.

Short answer

Question: Determine the domain and range for the function P(t) = 7.25t, which represents the total amount earned by an employee in a week when working t hours per week, given that the employee can work 4 to 6 days per week and 7 to 9 hours per day. Answer: The domain for the function is \(28 \leq t \leq 54\) and the range is \(203 \leq P(t) \leq 391.50\).

Step-by-step solution

Determine the constraints on the total work hours per week (t)

In the problem, we are told that the employee must work 4 to 6 days per week and 7 to 9 hours per day. Let's compute the minimum and maximum work hours per week using these constraints. Minimum work hours per week: 4 days/week * 7 hours/day = 28 hours/week Maximum work hours per week: 6 days/week * 9 hours/day = 54 hours/week Thus, the employee can work between 28 and 54 hours per week, inclusive.

Find the domain and range of the function P(t) = 7.25t

Now, we have the constraints on t, the total work hours per week. The domain can be described as the set of all possible input values for the function, that is, the total work hours per week. Domain: \(28 \leq t \leq 54\) Now let's find the corresponding range, which is the set of all possible output values for the function or the total amount earned per week. We can determine these values by plugging in the minimum and maximum values of t into the function P(t): Minimum amount earned per week: \(P(28) = 7.25(28) = \$203\) Maximum amount earned per week: \(P(54) = 7.25(54) = \$391.50\) Range: \(203 \leq P(t) \leq 391.50\)

Write the final answer

The domain and range for the given function P(t) are as follows: Domain: \(28 \leq t \leq 54\) Range: \(203 \leq P(t) \leq 391.50\)

Key concepts

Function Constraints

When dealing with functions, understanding the constraints is crucial. Constraints determine the limits or boundaries of the possible inputs and outputs for a function. In our exercise, the employee's workweek has specific constraints that affect the domain of the function that calculates their earnings.

• The employee must work between 4 to 6 days per week.
• Each workday must consist of 7 to 9 hours.

This implies a range of total hours worked per week, affecting which hours are permissible inputs for the function. For instance, if we calculate:

• *Minimum*: 4 days times 7 hours = 28 hours.
• *Maximum*: 6 days times 9 hours = 54 hours.

The constraints, therefore, dictate that the employee can work between 28 and 54 hours each week. This is crucial for defining the domain of the function.

Linear Functions

A linear function is one where the relationship between variables is constant. In simpler terms, this means that the rate of change is the same throughout the entire graph. In our scenario, the total earnings can be calculated using a linear function.

The function is represented as:\(P(t) = 7.25t\)
where \(t\) is the total hours worked each week, and \(7.25\) is the hourly rate.

Linear functions are easy to work with because they form straight lines when plotted on a graph. They follow the general formula:

\(y = mx + c\)
Where:

• \(y\) is the output or dependent variable (in our case, total earnings \(P(t)\))
• \(x\) is the input or independent variable (hours worked \(t\))
• \(m\) is the slope or rate of change ($7.25 per hour)
• \(c\) is the y-intercept (0 in this case, as there's no fixed starting point)

Understanding linear functions helps visualizing how small changes in input, such as hours worked, linearly affect the output, the total earnings.

Hourly Wage

The concept of hourly wage is straightforward but significant. It determines how much an employee earns for each hour worked. In our problem, the hourly wage is \(\\(7.25\). This figure is constant, meaning the employee receives this rate for every hour of work.

Let's look into the significance of knowing your hourly wage:

• **Earnings Calculation**: The hourly wage is pivotal for calculating weekly earnings. In our function, total earnings are obtained by multiplying the hours worked \( t \) by the hourly wage \( \\)7.25 \).
• **Budgeting**: An hourly wage helps employees plan their finances, understanding the minimum and maximum they can earn weekly.
• **Policy Setting**: For employers, setting an hourly wage influences scheduling and staffing. It may affect how many hours employees are encouraged to work.

Understanding your hourly wage helps illuminate how shifts in work hours can impact overall earnings, which is pivotal for financial planning.