Problem 42 - Calculus

Question

Which points between \(t=0\) and \(t=5\) can be in the domain of \(f(t)\) ? With this domain find the range in \(37-42\). $$ f(t)=2^{-t} $$

Step-by-Step Solution

Answer

Domain: [0, 5]; Range: [1/32, 1].

Step 1: Identify the Function

The function given is \(f(t) = 2^{-t}\). This function is defined for all real numbers \(t\) because it is an exponential function.

Step 2: Determine the Domain

The domain of \(f(t)\) is all real numbers. Since the question asks for points between \(t=0\) and \(t=5\), the domain in this context is \([0, 5]\).

Step 3: Determine the Range of \(f(t)\)

For \(t \in [0, 5]\), \(f(t)\) takes the form \(2^{-t}\). We calculate the smallest and largest values of \(f(t)\) in this interval to find the range.

Step 4: Evaluate \(f(t)\) at the Interval Endpoints

Calculate \(f(0) = 2^{-0} = 1\) and \(f(5) = 2^{-5} = \frac{1}{32}\). These calculations tell us that \(f(t)\) decreases monotonically from 1 to \(\frac{1}{32}\).

Step 5: Establish the Range for \(f(t)\)

Since \(f(t)\) is decreasing, the range of \(f(t)\) for \(t \in [0, 5]\) is from the maximum value 1 to the minimum value \(\frac{1}{32}\). Thus, the range is \([\frac{1}{32}, 1]\).

Key Concepts

Domain and Range

When discussing functions, the domain and range are essential components. The **domain** is the set of all possible input values (or 'inputs') for the function. For the function \( f(t) = 2^{-t} \), it can accept any real number as input, meaning its domain in a general sense is all real numbers \((-\infty, \infty)\). However, in specific contexts as given, limits like \([0, 5]\) can be set for practical reasons.

In the exercise, we limit the domain to \(t = 0\) to \(t = 5\), making it \([0, 5]\). The **range** of a function is the set of all possible output values. For the function \( f(t) = 2^{-t} \), the range depends on the interval of the domain. For \( t \in [0, 5] \), the outputs of the function decrease from 1 to \( \frac{1}{32} \), establishing the range as \([\frac{1}{32}, 1]\).

Understanding domain and range helps in predicting and explaining the behavior of functions, which is especially important in fields like mathematics, physics, and engineering.

Interval Notation

Interval notation is a convenient way to specify sets of numbers, especially domains and ranges in mathematics. It uses brackets to show the inclusivity of the endpoints.

• A square bracket \([\ ]\) indicates that the endpoint value is included in the interval.
• A parenthesis \((\ )\) indicates that the endpoint value is not included, representing an open interval.

For the domain \([0, 5]\), both 0 and 5 are included, meaning the function is defined and evaluated including these endpoints. Similarly, the range \([\frac{1}{32}, 1]\) includes both \(\frac{1}{32}\) and 1, showing that these are the extreme values \(f(t)\) can take when \(t\) is between 0 and 5.

Interval notation offers a precise, clear, and neat way to represent domains and ranges. It is a foundational skill that aids in exploring more complex algebraic and analytical tasks as one progresses in their academic journey.

Decreasing Functions

A decreasing function is one where the output values reduce as the input increases. For the function \( f(t) = 2^{-t} \), this property is clearly visible from the calculations.

To identify if a function is decreasing, we compare the values at different intervals. For instance, as \(t\) moves from 0 to 5:

• At \(t = 0\), \(f(t) = 1\)
• At \(t = 5\), \(f(t) = \frac{1}{32}\)

This indicates a downward movement in the function values, making it a decreasing function. Generally, with exponential functions like \(2^{-t}\), we observe that as \(t\) increases, \(f(t)\) decreases. Understanding decreasing functions is crucial in various applications, such as decay processes in physics and economics.