Problem 115 - Precalculus Essentials

Question

Furry Finances. A pet insurance policy has a monthly rate that is a function of the age of the insured dog or cat. For pets whose age does not exceed \(4,\) the monthly cost is \(\$ 20 .\) The cost then increases by \(\$ 2\) for each successive year of the pet's age. $$\begin{array}{cc} \text { Age Not Exceeding } & \text { Monthly cost } \\ \hline 4 & \$ 20 \\ 5 & \$ 22 \\ 6 & \$ 24 \end{array}$$ The cost schedule continues in this manner for ages not exceeding \(10 .\) The cost for pets whose ages exceed 10 is \(\$ 40 .\) Use this information to create a graph that shows the monthly cost of the insurance, \(f(x),\) for a pet of age \(x,\) where the function's domain is [0,14]

Step-by-Step Solution

Answer

The function would be defined as follows: \(f(x) = 20\) for \(0 \leq x \leq 4\), \(f(x) = 20 + 2(x - 4)\) for \(4 < x \leq 10\) and \(f(x) = 40\) for \(10 < x \leq 14\). The graph will be piecewise linear with jump discontinuities at x=4, x=10 and x=11.

Step 1: Define Function for Ages 0 to 4

For a pet whose age x does not exceed 4, the monthly cost of the insurance is a constant and it is $20. This can be written as \(f(x) = 20\) for \(0 \leq x \leq 4\).

Step 2: Define Function for Ages 5 to 10

For a pet with age x between 5 and 10, the monthly cost of the insurance increases with each year by $2, so the cost at age x is the cost at age 4 plus $2 for each extra year. This can be written as \(f(x) = 20 + 2(x - 4)\) for \(4 < x \leq 10\) .

Step 3: Define Function for Ages Exceeding 10

For a pet with age x exceeding 10, the monthly cost of the insurance is $40. This can be written as \(f(x) = 40\) for \(10 < x \leq 14\).

Step 4: Graphing the Function

Plot these three function definitions on the interval [0, 14]. Draw a filled dot at the points (4,20), (5,22), (10,30) and (11,40) because these values are included, while for x=4, x=10 and x=11 draw empty dots at the ends of the other function definitions to indicate exclusiveness.

Key Concepts

Domain and Range

When dealing with piecewise functions like in the pet insurance problem, understanding the domain and range is crucial. The **domain** refers to all the allowable input values for the function, in this case, the ages of the pets. Here, the domain is

• [0, 14], means a pet can be between 0 and 14 years old.

The **range** is all the possible output values or monthly costs that correspond to the domain. In this example, the range includes values such as:

• 20, 22, 24, ..., 30, and 40.

To identify these, think about how the cost changes:

• From 0 to 4 years, the cost is 20, so the range at this interval is just one value: 20.
• From over 4 to 10 years, the cost increases by 2 for each year over 4 until it reaches 30, making the range 20, 22, 24, ..., 30.
• Finally, for pets older than 10 up to 14, the cost is fixed at 40, thus only 40 is included here in the range.

Function Notation

Function notation offers a way to express relationships between inputs and outputs succinctly. This problem involves a function denoted by

• \( f(x) \).

The function notation tells us how monthly costs change based on pet age, \( x \). Let's break it down:

• For ages 0 to 4, the function is constant: \( f(x) = 20 \).
• For ages 5 to 10, the increment function takes over, growing linerally: \( f(x) = 20 + 2(x - 4) \). This indicates the base cost plus an additional \( 2(x - 4) \) for each year beyond four.
• For ages over 10, simplicity returns with \( f(x) = 40 \), a fixed cost for older pets.

Function notation like this makes it easy to compute how changes in age impact costs and helps in graphing these relationships.

Graphing Functions

Graphing functions provides a visual insight into how piecewise functions operate. Here, you need to create a piecewise graph that clearly shows different segments of ages and costs. Here's a step-by-step example of graphing:

• Start with the first segment, where \( f(x) = 20 \) from ages 0 to 4. This is a horizontal line at 20.
• Next, from ages 5 to 10, plot the increasing function \( f(x) = 20 + 2(x - 4) \). This line begins at 22 and increases until it reaches 30 at age 10.
• Lastly, plot the constant line \( f(x) = 40 \) for ages over 10 up to 14, representing the flat rate for older pets.

Remember to use filled dots to indicate inclusive values and hollow dots for exclusive values at transitions, like at (4,20) or (10,30), highlighting that the function jumps at those points. Graphing functions like these helps you quickly interpret the relationship between pet age and insurance costs.