Question

A function \(d(t)\) is being used to predict the distance a bird flies \(t\) hours after release from a wildlife rehab program. What numbers are acceptable domain values? A. any number will do B. only positive numbers C. only integers D. only non-negative numbers

Short answer

D. only non-negative numbers

Step-by-step solution

Understand the Context

The function \(d(t)\) gives the distance a bird flies after \(t\) hours. The value of \(t\) represents the time in hours.

Consider the Meaning of Time

Time \(t\) is measured in hours and must make sense in a real-world context, so \(t\) should be a relevant unit of measurement for time.

Analyze Possible Domain Values

Time cannot be negative because it represents a duration starting from the release of the bird. Therefore, \(t\) must be non-negative.

Select the Appropriate Domain

Since time can be 0 hours (initial release) or any positive value, the acceptable domain values for \(t\) are non-negative numbers.

Key concepts

domain and range

In mathematics, functions are rules that relate inputs to outputs. The set of all possible inputs for a function is called its domain. For the function \(d(t)\), which predicts the distance a bird flies after \(t\) hours, the domain consists of all the valid values of \(t\).
Since \(t\) represents time, and time cannot be negative in real-world scenarios, the domain of \(d(t)\) is all non-negative numbers. This means that \(t\) can be 0 (the moment of release) or any positive number representing the hours after release.

Understanding the domain is crucial. It ensures that we use the function in a way that makes sense.
In this case, because the function deals with time, and time progresses in a positive direction, our domain must exclude negative values.

Now, let's think about the range. The range is the set of all possible outputs of the function. For \(d(t)\), it's the set of distances the bird can fly. Since distance cannot be negative either, the range of \(d(t)\) will also be non-negative numbers.

real-world applications

Functions, like \(d(t)\), are not just abstract concepts. They have practical uses in the real world. Here, \(d(t)\) helps us predict how far a bird can fly after its release. This can be essential for wildlife rehabilitators to track the bird's progress and ensure it's adjusting well to the wild.

In ecology and conservation, similar functions can be used to:

• Predict population growth over time.
• Estimate the spread of a species in a new habitat.
• Track migration patterns.

These functions help scientists make informed decisions and create better conservation strategies.
They use mathematical models to simulate and study wildlife behavior, ensuring that species can thrive in their natural habitats.

Moreover, functions are used in various fields like:

• Economics to predict market trends
• Engineering to design systems and structures
• Medicine to model the spread of diseases

Understanding how functions operate can provide powerful insights and useful predictions applicable to many areas of our lives.

time measurement

Time is an essential variable in many functions, including \(d(t)\), where \(t\) represents time in hours after the bird's release. In mathematics, time is typically considered a continuous variable. This means it can take on any value within a specified interval, rather than just discrete values like whole numbers.

Measuring time accurately is crucial because it impacts the function's accuracy. For example:

• If we measured \(t\) in days instead of hours, the function \(d(t)\) would need to adjust accordingly.
• The domain would change to non-negative days.
• The predictions on the bird's flight distance would vary.

Time measurement needs to be clear and consistent to maintain the function's reliability.

In real-world scenarios, different contexts might call for different units of time, such as:

• Seconds (in split-second decisions like sports)
• Minutes (for shorter events)
• Days or years (for long-term trends or studies)

For students, understanding how time functions as a variable helps in analyzing data and interpreting results accurately in various scientific and mathematical contexts.