91Ó°ÊÓ

The domain and range of a function are fundamental concepts in mathematics. The **domain** refers to all the possible input values (or x-values) for which the function is defined. In this exercise, the function that approximates the age of a female blue whale in terms of its length has a domain given by \(24 < x < 87.\)
This means that the function only works for whale lengths between 24 feet and 87 feet. Why? Because these values ensure that the argument of the logarithm, \(\frac{87 - x}{63},\) is positive. A logarithm function is only defined for positive arguments.
So here, \(87 - x\) must always be greater than zero, therefore ensuring that \(x < 87\). Additionally, to keep the marine biology context realistic, the length \(x\) must also be above 24 feet because smaller values aren’t typical for blue whales.
The **range** of a function represents all possible output values (or y-values) after applying the function. In this case, it concerns the age in years that we get after plugging different x-values into our function.

The natural logarithm (ln) is a special type of logarithm where the base is the mathematical constant *e* (\text{approximately, 2.71828}). It is denoted as \(\text{ln}(x)\).
In our function, the natural logarithm helps to create a more accurate model for approximating the age of a whale based on its length. The properties of natural logarithms make them particularly useful.
* For instance, one property used is \(\text{ln}\big(\frac{1}{a}\big) = -\text{ln}(a)\), which can be seen in the exercise. \text{\(\text{ln}\big(\frac{1}{a}\big)\)} means we take the reciprocal of *a* which inverts the factor, transforming the log into a negative value.
Knowing how to manipulate and calculate natural logarithms, either through a calculator or logarithm tables, is key to solving problems like these efficiently.

Approximating the age of a whale by using logarithmic functions highlights an interesting aspect of natural sciences and applied mathematics. The given function \(f(x) = -2.57 \text{ln}\big(\frac{87 - x}{63}\big)\) links the length of the whale (input) to its age (output).
To solve for age, let’s break down how it works:

* Substitute the whale’s length \(x\) into the function.
* Simplify the fraction inside the logarithm.
* Apply properties of logarithms to simplify further.
* Calculate the ln value using \(\text{ln}(x)\) calculators or tables.
* Multiply the result by the coefficient (-2.57 in this case).

**Example**: For a whale of 80 feet in length:
* Substituting \(80\) for \(x\), we find \(f(80) = -2.57 \text{ln}\big(\frac{87 - 80}{63}\big)\) simplifies to \(f(80) = 2.57 \text{ln}(9)\).
* Calculating \(\text{ln}(9)\) results in approximately 2.197.
* Therefore, \(2.57 \times 2.197 \backsimeq 5.65\).
The whale is approximately 5.65 years old. This approximation allows scientists to make informed guesses about a whale’s age based on observable attributes like length.