Problem 22 - Applied Calculus

Question

For each function: a. Evaluate the given expression. b. Find the domain of the function. c. Find the range. $$ f(x)=-\sqrt{-x} ; \text { find } f(-100) $$

Step-by-Step Solution

Answer

f(-100) = -10; Domain: \((-\infty, 0]\); Range: \((-\infty, 0]\).

Step 1: Evaluate the Expression

First, substituting \(x = -100\) into the function \(f(x) = -\sqrt{-x}\), we have: \[ f(-100) = -\sqrt{-(-100)} \]}{

Step 2: Simplify the Expression

Now, simplify the expression inside the square root: \(-(-100) = 100\). So, \[ f(-100) = -\sqrt{100} \] \[ f(-100) = -10 \]

Step 3: Determine the Domain of the Function

The domain of a function involving a square root, like \(-\sqrt{-x}\), is determined by setting the expression inside the square root greater than or equal to 0: \(-x \geq 0\) Therefore, \(x \leq 0\). So, the domain of the function is \(x \in (-\infty, 0]\).

Step 4: Determine the Range of the Function

Since \(-\sqrt{-x}\) produces negative values and zero, the range consists of all non-positive numbers. The square root will cover all non-negative real numbers, but due to the negative sign in front of the square root, the range is \((-\infty, 0]\).

Key Concepts

Square Root Functions

When we talk about square root functions, we are dealing with functions that involve the square root symbol (√). These functions will produce outputs by extracting the square root of a given input, usually a non-negative number.
In mathematics, we commonly see the square root function written as \(\sqrt{x}\). It simplifies to the value that, when multiplied by itself, gives back the original number \(x\). For example, \(\sqrt{4} = 2\) because \(2 \times 2 = 4\).
However, situations change when the expression involves more complexity, such as negative signs or other operations within the square root. In the context of the problem we are discussing, we have \(-\sqrt{-x}\). This differs from the simple square root of \(x\) by involving a negative input, allowing negative values in a very particular manner.

• This function, \(-\sqrt{-x}\), accepts non-positive inputs.
• The negative sign outside the square root affects the output, flipping the sign of the result.

Understanding these nuances is key to working reliably with square root functions in different forms. Feel free to try substituting various numbers to see how it changes the output! Remember always to handle the expression inside the square root carefully to know what values are permissible for input.

Function Evaluation

Function evaluation is a simple but crucial concept in math. It involves replacing a variable within a function with a given number and then simplifying it to find the result.
Consider the function \(f(x) = -\sqrt{-x}\). To find \(f(-100)\), you substitute \(x = -100\) directly into the function.
Here's how it works step-by-step:

• First, replace \(x\) with \(-100\): \(-\sqrt{-(-100)}\).
• Then, simplify the term inside the square root: \(-(-100) = 100\).
• Finally, compute the square root and apply the negative sign: \(-\sqrt{100} = -10\).

Thus, when \(x = -100\), the function evaluates to \(-10\). Function evaluation helps us understand what output we can expect from giving specific inputs. It's useful across many areas of math, including finding particular characteristics like the domain and range of functions.

Negative Numbers in Functions

Negative numbers present interesting challenges and features in mathematical functions. Specifically, in our context with \(-\sqrt{-x}\), understanding how negative numbers affect calculations is essential.
Consider the expression \(-\sqrt{-x}\). Here, two negatives interact:

• The negative sign inside the square root ensures that \(x\) must be non-positive (zero or less).
• The negative sign outside the square root then reverses the sign of the output of \(\sqrt{-x}\) after it's been solved.

Negative numbers by themselves imply the opposite of their positive counterparts. However, when employed in mathematical operations like our square root scenario, they enable different solutions and results.
Therefore, working with negative numbers inside functions can dramatically alter what inputs are valid, impacting both the domain and range of the function. It's important always to consider the location of the negative signs in the expression to predict the behavior of the function accurately.