Problem 19 - Applied Calculus

Question

For each function: a. Evaluate the given expression. b. Find the domain of the function. c. Find the range. $$ f(x)=\sqrt{4-x^{2}} ; \text { find } f(0) $$

Step-by-Step Solution

Answer

a) \( f(0) = 2 \); b) Domain: \([-2, 2]\); c) Range: \([0, 2]\).

Step 1: Evaluate the Expression at Given Point

To evaluate \( f(0) \), substitute \( x = 0 \) into the function: \( f(x) = \sqrt{4 - x^2} \). So, \( f(0) = \sqrt{4 - 0^2} = \sqrt{4} = 2 \).

Step 2: Find the Domain of the Function

The expression inside the square root, \( 4 - x^2 \), must be greater than or equal to zero for \( f(x) \) to be defined. Hence, we set up the inequality: \( 4 - x^2 \geq 0 \). This simplifies to \( x^2 \leq 4 \), or \( -2 \leq x \leq 2 \). Thus, the domain of the function is \([-2, 2]\).

Step 3: Find the Range of the Function

Since the expression inside the square root can vary between 0 and 4, we calculate the corresponding values of \( f(x) = \sqrt{4 - x^2} \). When \( x = 0 \), \( f(x) = 2 \); when \( x = \pm 2 \), \( f(x) = 0 \). Therefore, as \( x \) varies from \(-2\) to \(2\), \( f(x) \) varies from 0 to 2. Hence, the range of \( f(x) \) is \([0, 2]\).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (usually \( x \) values) that allow the function to work without any mathematical issues. For functions that include square roots, such as \( f(x) = \sqrt{4 - x^2} \), you need to make sure what's inside the square root stays non-negative.
For this function, we check when \( 4 - x^2 \geq 0 \) because square roots of negative numbers aren't real.
Solving \( 4 - x^2 \geq 0 \) gives us \( x^2 \leq 4 \), which further simplifies to \( -2 \leq x \leq 2 \).
Thus, the domain of this function is the set of \( x \) values from -2 to 2, inclusive.

• Always ensure the inside of a square root is non-negative.
• Domains keep functions defined and real.

Range of a Function

The range of a function is the set of all possible output values (\( y \) values) it can produce. Once we know the domain, we can find the range by examining the outputs based on the allowed inputs.
For \( f(x) = \sqrt{4 - x^2} \), the output values are influenced directly by what’s under the square root.
When \( x = 0 \), \( f(x) = 2 \), and when \( x = \pm 2 \), \( f(x) = 0 \). These values represent the extremes of the function as \( x \) varies from -2 to 2.
So, the function's range is \([0, 2]\).

• Range represents all possible outputs given the domain.
• Consider maximum and minimum values within the domain.

Evaluating Functions

Evaluating a function simply means finding the output for a given input. This involves substituting a specific input value into the function and simplifying.
Let’s evaluate \( f(0) \) in the function \( f(x) = \sqrt{4 - x^2} \).

• Substitute \( x = 0 \) into the function: \( f(x) = \sqrt{4 - 0^2} = \sqrt{4} = 2 \).
• This gives us the output \( f(0) = 2 \).

Understanding this helps in interpreting function behavior for individual points.

• Substitute and calculate step-by-step.
• Clear arithmetic helps avoid mistakes.