Problem 4 - Thomas Calculus

Question

In Exercises \(1-6,\) find the domain and range of each function. $$ F(t)=\frac{1}{1+\sqrt{t}} $$

Step-by-Step Solution

Answer

Domain: \( t \geq 0 \), Range: \( 0 < F(t) \leq 1 \).

Step 1: Identify the Domain

The domain of a function is the set of all possible input values (in this case, values of \( t \)) for which the function is defined. For the function \( F(t) = \frac{1}{1 + \sqrt{t}} \), the expression under the square root, \( t \), must be non-negative because the square root of a negative number is not defined in the real number system. Therefore, \( t \geq 0 \). Additionally, the denominator \( 1 + \sqrt{t} eq 0 \). However, since \( \sqrt{t} \geq 0 \), the denominator will never be zero. Thus, the domain of \( F(t) \) is all non-negative real numbers: \( t \geq 0 \).

Step 2: Determine the Range

The range of a function is the set of all possible output values. For \( F(t) = \frac{1}{1 + \sqrt{t}} \), we evaluate the behavior of the function as \( t \to 0 \) and as \( t \to \infty \). When \( t = 0 \), \( F(t) \) approximates to \( \frac{1}{1+0} = 1 \). As \( t \to \infty \), \( \sqrt{t} \to \infty \), so \( 1 + \sqrt{t} \to \infty \), and consequently, \( F(t) \to 0 \). Thus, the range of \( F(t) \) is all values \( 0 < F(t) \leq 1 \).

Key Concepts

Function Analysis

Function analysis is an essential step in understanding how functions behave. It involves determining important features like the domain and range. In the context of functions, the "domain" refers to all the possible input values that a function can accept. For example, a function with a square root requires its input to be non-negative, because you cannot take the square root of a negative number in the real number system. The "range" is the set of possible output values that the function can produce.

Analyzing a function like \( F(t) = \frac{1}{1+\sqrt{t}} \), we start by determining allowable inputs—this is the domain. Next, we look at the outputs—this is the range. During this analysis, you check conditions that the function should respect to ensure it behaves consistently.

• Domain: Allows for legitimate function computation.
• Range: Shows potential outcomes of the function.

Remember, comprehending these aspects is vital for solving exercises accurately.

Square Root Function

Square root functions are special because they only work with non-negative numbers. Their basic form is \( \sqrt{x} \). This function increases as the input value increases, which is why it is important to carefully consider domain and range in these functions.

In a composite function like \( F(t) = \frac{1}{1 + \sqrt{t}} \), the square root function affects the denominator, making the value under the root, \( t \), dictate its domain. \( t \) must be greater than or equal to zero to avoid any math errors. The impact of \( \sqrt{t} \) on the function highlights:

• The function can handle only inputs from \([0, \infty)\).
• It demonstrates a continuous growth in value without turning negative.

When working with square root functions, always examine the values under the root to understand key limits and behaviors.

Real Number System

The real number system includes all the numbers we utilize in everyday math, such as integers, fractions, and decimals. When doing function analysis, particularly involving functions like \( F(t) = \frac{1}{1+\sqrt{t}} \), it's crucial to consider how operations are defined under this system.

For instance, within the real number system, taking the square root of a negative number is undefined. Hence, the domain of any square root function is restricted to non-negative numbers. Understanding these restrictions prevents errors in function computation.

• Always check if a value is suitable under real numbers.
• The real number system forms the basis for determining function limits.

Knowing these boundaries helps you to define where the function is applicable, which is critical in solving math problems efficiently.