Problem 34 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=\sqrt{-1-x}$$

Step-by-Step Solution

Answer

Graph is an upward curve starting at (-1, 0) and rising as x decreases.

Step 1: Identify the Domain of the Function

The function given is \( f(x) = \sqrt{-1-x} \). First, we must determine the domain of the function. The expression inside the square root, \(-1-x\), must be greater than or equal to zero for the function to be real (since you cannot take the square root of a negative number). Thus, we solve the inequality: \(-1-x \geq 0\). This simplifies to \(-x \geq 1\). Multiplying by -1 gives \(x \leq -1\). Therefore, the domain of the function is \(x \leq -1\).

Step 2: Identify and Evaluate Key Points

Using the domain \(x \leq -1\), evaluate the function at key points to help graph it. Begin with the boundary point \(x = -1\): \(f(-1) = \sqrt{-1-(-1)} = \sqrt{0} = 0\). Next, choose another point within the domain, such as \(x = -2\): \(f(-2) = \sqrt{-1-(-2)} = \sqrt{1} = 1\). Check more negative values to see how the function behaves as \(x\) decreases.

Step 3: Analyze the Behavior

Consider the behavior of the function as \(x\) takes values less than -1. Notice that as \(x\) becomes more negative, \(-1-x\) becomes larger positive, hence \(f(x)\) increases. For example, \(f(-3) = \sqrt{-1-(-3)}=\sqrt{2}\). This indicates that the function value becomes larger as \(x\) decreases.

Step 4: Sketch the Graph

With the evaluated points and considering the behavior of the function, sketch the graph. Draw an arc starting at \((-1, 0)\) that moves upwards as x decreases. The graph shows that \(f(x)\) increases slowly and smoothly as x goes to more negative values. The domain restriction \(x \leq -1\) means there is no part of the graph for \(x > -1\).

Step 5: State the Range

The range is determined by the values that \(f(x)\) can take. Since \(f(x) = \sqrt{-1-x} \) begins at 0 when \(x = -1\) and increases as \(x\) decreases, the range is \(f(x) \geq 0\).

Key Concepts

Understanding Function Domain

When graphing a function like \( f(x) = \sqrt{-1-x} \), one of the first steps is to determine the domain of the function. The domain comprises all the possible input values \( x \) for which the function is defined. In this case, we are dealing with a square root function, which requires the expression under the square root to be non-negative. Therefore, we set the inequality \(-1-x \geq 0\) to find the valid values for \( x \).
Re-arranging the inequality gives

• \(-x \geq 1\)
• \(x \leq -1\) (by multiplying both sides by -1 and remembering to flip the inequality sign)

You can now conclude that the domain of \( f(x) \) is all real numbers less than or equal to -1. Knowing the domain is crucial, as it tells us where on the x-axis we should plot our function.

Graphing Square Root Functions

Graphing square root functions involves a few steps to get an accurate picture of the curve. For the function \( f(x) = \sqrt{-1-x} \), we know that its domain is \( x \leq -1 \), so our graph will be restricted to this interval.

We start by evaluating the function at key points within the domain. Here’s how:

• At \( x = -1 \): \( f(-1) = \sqrt{-1-(-1)} = \sqrt{0} = 0 \)
• At \( x = -2 \): \( f(-2) = \sqrt{-1-(-2)} = \sqrt{1} = 1 \)
• At \( x = -3 \): \( f(-3) = \sqrt{-1-(-3)} = \sqrt{2} \approx 1.41 \)

Plotting these calculated points helps in visualizing the curve. As you plot these points, draw a smooth, upward curving path starting from \((-1, 0)\) that crosses through these points as \( x \) becomes more negative. This illustrates how the value of \( f(x) \) increases as \( x \) decreases.

Exploring the Function Range

The range of a function encompasses all the possible output values \( f(x) \) can take. For \( f(x) = \sqrt{-1-x} \), the function begins at \( x = -1 \) where the output is \( f(x) = 0 \). As \( x \) continues to decrease, the value of the expression inside the square root becomes larger, increasing the value of \( f(x) \).

To summarize, since our starting output point is 0 and moves upwards as we pick more negative \( x \) values (like \( x = -2, -3,... \)), the range of \( f(x) \) is all real numbers greater than or equal to 0. Mathematically, we can express this as

• The range is \( f(x) \geq 0 \).

This means that as you graph these functions, you'll see outputs that start at 0 and extend upwards. Understanding the range before graphing allows you to immediately know the vertical extent of your graph.