Problem 33 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=\sqrt{2-x}$$

Step-by-Step Solution

Answer

Graph \(f(x) = \sqrt{2-x}\) from \(x = 2\) leftward, plotting key values to form a decreasing curve.

Step 1: Understand the Function

The function given is \(f(x) = \sqrt{2-x}\). This is a square root function. The expression under the square root must be non-negative for real number outputs. Therefore, we need the domain where \(2-x \geq 0\).

Step 2: Find the Domain of the Function

To find the domain, solve the inequality \(2-x \geq 0\). Rearranging gives \(x \leq 2\). Thus, the domain of the function is \((-\infty, 2]\).

Step 3: Plot Key Points

Select key values from the domain to evaluate the function: \(x = 2, 1, 0, -1\). Calculate \(f(2) = \sqrt{0} = 0\), \(f(1) = \sqrt{1} = 1\), \(f(0) = \sqrt{2} \approx 1.41\), and \(f(-1) = \sqrt{3} \approx 1.73\).

Step 4: Consider the Function's Behavior

The function \(f(x) = \sqrt{2-x}\) is decreasing since as \(x\) increases towards 2 from the left, the value under the square root decreases and approaches zero. This means the graph will slope downward as \(x\) moves right towards 2.

Step 5: Sketch the Graph

Plot the calculated points (2, 0), (1, 1), (0, 1.41), and (-1, 1.73) on the coordinate plane. Connect these points with a smooth curve starting from \(x = 2\) and moving left. The curve should start at (2,0) and rise as \(x\) decreases, staying above the x-axis and never dropping below 0.

Key Concepts

Domain of Functions

When dealing with square root functions like \( f(x) = \sqrt{2-x} \), understanding the domain is crucial. The domain tells you which values of \( x \) can be plugged into the function without causing problems like taking the square root of a negative number. For the given function, we need to find where the expression inside the square root is non-negative, meaning \( 2-x \geq 0 \).
Rearranging this inequality gives \( x \leq 2 \). Therefore, the domain of \( f(x) = \sqrt{2-x} \) is \( (-\infty, 2] \).

• The expression \(2-x\) must be positive or zero for the function to be defined.
• As a result, \( x \) can be any number less than or equal to 2.

This domain information is essential for graphing, as it dictates where the graph will exist on the \( x \)-axis.

Plotting Key Points

After defining the domain, the next step in graphing a square root function is to identify key points to plot. These are specific \( x \) values within the domain that help outline the graph’s shape. For \( f(x) = \sqrt{2-x} \), choose values of \( x \) such as 2, 1, 0, and -1.
Calculating for these values:

• \( f(2) = \sqrt{0} = 0 \)
• \( f(1) = \sqrt{1} = 1 \)
• \( f(0) = \sqrt{2} \approx 1.41 \)
• \( f(-1) = \sqrt{3} \approx 1.73 \)

These points \((2, 0), (1, 1), (0, 1.41), (-1, 1.73)\) are plotted on the coordinate plane and connected smoothly. Choosing the right points allows you to sketch the curve accurately and understand the rise or fall of the graph.

Function Behavior Analysis

Analyzing how the function behaves across its domain gives insights into its general shape and direction. For \( f(x) = \sqrt{2-x} \), one key characteristic is that it decreases as \( x \) increases towards 2. This means that as you move right along the \( x \)-axis, the output values decrease.

• When \( x \) is at its maximum value in the domain, which is 2, the output is 0 (since \( \sqrt{0} = 0 \)).
• As \( x \) decreases from 2, the function value increases, since the expression under the square root and consequently the square root itself, increases.

This behavior gives the graph a downward slope as \( x \) approaches 2 from the left. Understanding this behavior is crucial for accurately sketching the overall graph and predicting function trends outside the explicitly calculated key points.