Problem 21 - Applied Calculus

Question

For each function: a. Evaluate the given expression. b. Find the domain of the function. c. Find the range. $$ f(x)=\sqrt{-x} ; \text { find } f(-25) $$

Step-by-Step Solution

Answer

f(-25) = 5, Domain: \((-\infty, 0]\), Range: \([0, \infty)\).

Step 1: Evaluate the Expression

To find \( f(-25) \), substitute \(-25\) into the function: \[ f(-25) = \sqrt{-(-25)} = \sqrt{25} = 5 \] Thus, \( f(-25) = 5 \).

Step 2: Identify the Domain

The domain of the function \( f(x) = \sqrt{-x} \) is determined by the values of \( x \) for which the expression under the square root is non-negative. Since \( -x \geq 0 \), it follows that \( x \leq 0 \). Therefore, the domain is all non-positive numbers, or \((-\infty, 0]\).

Step 3: Determine the Range

The range of \( f(x) = \sqrt{-x} \) is related to the outputs of the function. Since the square root function returns non-negative values, \( f(x) \geq 0 \). Therefore, the range is all non-negative numbers, or \([0, \infty)\).

Key Concepts

Square Root Function

A square root function, like the one given by \( f(x) = \sqrt{-x} \), involves taking the square root of a variable expression. The main goal is to find values of \( x \) that make the expression under the square root non-negative, ensuring a real number output. This means we need to make sure the value inside the square root is zero or positive to have real-valued results.
- In our function, since it's \( \sqrt{-x} \), it's crucial that \( -x \geq 0 \). This means \( x \leq 0 \). Therefore: - The domain consists of values of \( x \) that keep the expression under the square root non-negative. - In this context, we're considering non-positive values for \( x \). This results in a domain of \(( -\infty, 0 ] \).
Understanding these principles is vital when working with square root functions, as incorrect assumptions about the domain can lead to invalid results.

Evaluating Expressions

Evaluating an expression like \( f(x) = \sqrt{-x} \) involves substituting a value into the function and simplifying. To evaluate \( f(-25) \):
1. Substitute \(-25\) for \( x \) in the function: \( \sqrt{-(-25)} \).2. Simplify the expression inside the square root: \( \sqrt{25} \).3. Finally, simplify \( \sqrt{25} \) to obtain the result \( 5 \).
Therefore, \( f(-25) = 5 \). Through evaluation, we understand how input values transform into output values. This practice helps in verifying certain values lie within the domain and behave correctly according to the function's rules.

Function Analysis

Function analysis entails understanding both the domain and range, helping us comprehend the overall behavior of the function.
- **Domain:** This defines all permissible input values for which the function is defined. For the square root function \( f(x) = \sqrt{-x} \), we concluded earlier that the domain is \( x \leq 0 \).- **Range:** This defines all possible output values the function can produce. For our function, since \( \sqrt{-x} \) always returns non-negative values, the range is \([0, \infty)\).
This comprehensive analysis adapts the basic principles of square root behavior to unique functions. Grasping the relationship between domain and range is key to predicting a function's output given any particular input. This knowledge about function behavior supports further studies in calculus or advanced algebra, where these foundational principles are often utilized.