Problem 6 - Thomas Calculus

Question

In Exercises \(1-6,\) find the domain and range of each function. $$ g(z)=\frac{1}{\sqrt{4-z^{2}}} $$

Step-by-Step Solution

Answer

The domain is \((-2, 2)\) and the range is \((0, \infty)\).

Step 1: Understanding the Domain Conditions

The function is defined as \( g(z) = \frac{1}{\sqrt{4-z^{2}}} \). For the function to be defined, the expression under the square root \( 4 - z^2 \) must be greater than zero. This is because the square root of a negative number is not defined in the real numbers and division by zero is not allowed. Therefore, we need to solve the inequality: \( 4 - z^2 > 0 \).

Step 2: Solving for Domain

To solve \( 4 - z^2 > 0 \), we rewrite the inequality as \( 4 > z^2 \), which further simplifies to \(-2 < z < 2\). Hence, the domain of \( g(z) \) is the interval \((-2, 2)\).

Step 3: Determining the Range

Since the denominator \( \sqrt{4-z^2} \) must always be positive within the domain, \( g(z) = \frac{1}{\sqrt{4-z^2}} \) will output positive values. As \( z \) approaches \(-2\) or \(2\), \( \sqrt{4-z^2} \) approaches \(0\), making \( g(z) \) increase indefinitely. Therefore, the range of \( g(z) \) is \((0, \infty)\), as \( g(z) \) can take any positive value.

Key Concepts

Function Analysis

Analyzing functions means exploring the behavior and characteristics of mathematical expressions. In this case, we are looking at the function \( g(z) = \frac{1}{\sqrt{4-z^2}} \). When performing a function analysis, we often start by identifying the constraints on the function's domain and range.

- **Domain**: This refers to all the possible input values \( z \) for which the function is defined. We must keep in mind any mathematical operations that could restrict this, such as division by zero or taking the square root of a negative number.- **Range**: This encompasses the set of possible output values of the function. Depending on the behavior of the function, this can vary significantly.

By examining these components, we determine where the function exists and what outputs it can generate, which is crucial for graphs and applications.

Inequalities

Inequalities are essential in solving for domains and ensuring mathematical expressions fall within acceptable ranges. For the function \( g(z) = \frac{1}{\sqrt{4-z^2}} \), the inequality \( 4 - z^2 > 0 \) is pivotal.

Here's why this inequality matters:

• It ensures non-negative numbers under the square root.
• It avoids division by zero, which is undefined.

To solve \( 4 - z^2 > 0 \), we rearrange it to \( -2 < z < 2 \). This result indicates that only values between \(-2\) and \(2\) satisfy the equation, establishing the domain. Such inequalities help define the viable inputs, allowing functions to perform correctly.

Square Roots

Square roots are a fundamental concept often appearing in functions, especially where the input's square impacts computation. In \( g(z) = \frac{1}{\sqrt{4-z^2}} \), the square root affects both domain and range.

- **Defining the Domain**: A square root demands its input to be zero or positive. Thus, \( 4 - z^2 > 0 \) guides our choice of \( z \).- **Range Considerations**: When a function includes a reciprocal square root, as in our function, it affects how output values behave. As \( z \) approaches extremes of its domain, the denominator of \( g(z) \) approaches zero, shooting the values of \( g(z) \) toward infinity. Hence, the range becomes \( (0, \infty) \).

Understanding how square roots and their inverses operate within a function can illuminate broader mathematical principles, impacting both practical calculations and theoretical exploration.