Problem 5 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

Domain: \([-2, 2]\); Range: \([0, 2]\).

Step 1: Understand the function

The function given is \( g(z) = \sqrt{4 - z^2} \). This is a square root function, which means the expression inside the square root, \( 4 - z^2 \), must be non-negative for \( g(z) \) to be real-valued.

Step 2: Determine the domain

The domain of the function specifies the allowable values of \( z \). We need to solve the inequality \( 4 - z^2 \geq 0 \) to find the domain.Starting with the equation:\[ 4 - z^2 \geq 0 \]Rearrange this to:\[ z^2 \leq 4 \]Taking the square root on both sides, we find:\[ -2 \leq z \leq 2 \]So the domain of \( g(z) \) is \( z \in [-2, 2] \).

Step 3: Determine the range

Since \( g(z) = \sqrt{4-z^2} \), it can achieve a minimum value of 0 when \( 4-z^2 = 0 \) (i.e., when \( z = \pm 2 \)), and a maximum value when \( 4-z^2 \) is maximized. The maximum occurs when \( z = 0 \), resulting in \( \sqrt{4} = 2 \).Therefore, the range of \( g(z) \) is \( [0, 2] \).

Key Concepts

Square Root Functions

Square root functions are a type of function where the variable appears inside a square root symbol, like in the expression \( g(z) = \sqrt{4 - z^2} \). The key aspect of square root functions is that the expression under the root, called the radicand, must be non-negative. This ensures the function produces real numbers; otherwise, the function would output complex numbers.

For the function \( g(z) = \sqrt{4 - z^2} \):

• The radicand is \( 4 - z^2 \). It must be greater than or equal to zero.
• This condition limits the values that \( z \) can take, defining the domain of the function.

Understanding square root functions help in solving inequalities and finding the domain and range effectively. This allows us to analyze what values make the function valid and which outputs can result from those inputs.

Inequalities

Inequalities are mathematical statements that define a range of possible values for variables. When working with square root functions like \( g(z) = \sqrt{4 - z^2} \), inequalities play a crucial role in finding the domain.

To ensure \( 4 - z^2 \) is non-negative, set up the inequality:

• \( 4 - z^2 \geq 0 \)

Rearrange to:

• \( z^2 \leq 4 \)

Taking the square root gives:

• \(-2 \leq z \leq 2 \)

Thus, inequalities show that \( z \) must lie between \(-2\) and \(2\). This inequality is essential in determining the domain. Grasping how to solve inequalities is valuable for a variety of mathematical problems, particularly when involving functions like these.

Real-Valued Functions

Real-valued functions are functions that produce outputs which are real numbers. For square root functions like \( g(z) = \sqrt{4 - z^2} \), we must ensure that the output is always a real number.

In order to get real values for \( g(z) \):

• The radicand \( 4 - z^2 \) must be non-negative.

By managing the domain correctly, where \( z \) values are between \(-2\) and \(2\), \( g(z) \) can only provide real-number results. Accordingly, the range of the function, determined by evaluating possible outputs, turns out to be \([0, 2]\).

Recognizing real-valued functions helps to properly address which outputs are valid, avoiding the pitfalls of complex number results unless explicitly wanted. This precision ensures mathematical expressions like these function clearly within real-world or applied math contexts.