Problem 59 - Algebra 2

Question

Graph each function. State the domain and range. $$ y=\sqrt{x}-1 $$

Step-by-Step Solution

Answer

Domain: \( x \geq 0 \); Range: \( y \geq -1 \).

Step 1: Understand the Function

The function given is \( y = \sqrt{x} - 1 \). This is a transformation of the basic square root function \( y = \sqrt{x} \), where the function is shifted downward by 1 unit.

Step 2: Identify the Domain

The domain of the function is determined by the values of \( x \) for which the expression under the square root is non-negative. For \( \sqrt{x} \), \( x \) must be greater than or equal to 0. Hence, the domain of \( y = \sqrt{x} - 1 \) is all non-negative numbers: \( x \geq 0 \).

Step 3: Determine the Range

The range is determined by the possible values of \( y \). As \( x \) starts at 0 and increases, \( \sqrt{x} \) ranges from 0 to infinity. Therefore, \( y = \sqrt{x} - 1 \) ranges from \( -1 \) to infinity, because subtracting 1 from the smallest output (0) gives \(-1\). So, the range is \( y \geq -1 \).

Step 4: Plot Key Points

To graph the function, calculate a few key points: \( (0, -1) \), \( (1, 0) \), \( (4, 1) \), and \( (9, 2) \). These points are calculated by substituting values of \( x \) into the function.

Step 5: Sketch the Graph

Plot the key points on a coordinate grid and draw a smooth curve that passes through them, beginning at the point \( (0, -1) \) and moving upward to the right.

Key Concepts

Understanding Domain and Range

The domain of a function is crucial in understanding where the function is defined. For the function \( y = \sqrt{x} - 1 \), we focus on the expression inside the square root. It's essential that this expression, \( \sqrt{x} \), is defined for non-negative values only, meaning \( x \geq 0 \). This is because the square root of a negative number is not real in the set of real numbers. Therefore, the domain of this function is all points where \( x \) is greater than or equal to 0.

To determine the range, consider the smallest possible output of the square root function, which is 0, and subtract 1. This means the smallest value that \( y \) can take is \(-1\). As \( x \) grows larger, the square root \( \sqrt{x} \) also increases without bound. Hence, the range is \( y \geq -1 \).

Remember:

• Domain: \( x \geq 0 \)
• Range: \( y \geq -1 \)

Function Transformation Essentials

Function transformations modify a basic function to alter its graph. In this case, \( y = \sqrt{x} - 1 \) is a vertical shift of the basic square root function \( y = \sqrt{x} \).

The original function, \( y = \sqrt{x} \), is transformed by moving it directly down by 1 unit on the coordinate plane. This "down" movement is represented by subtracting 1 outside the square root notation. It's important to note that horizontal movements would affect the \( x \) inside the square root.

Here’s how transformations work:

• Vertical shifts: Adding or subtracting outside the function (\( \pm C\)).
• Horizontal shifts: Affecting the \( x \) inside the function (like \( \sqrt{x + C} \)).
• Reflection and stretching/shrinking can also occur, but they are not part of this exercise.

Coordinate Graphing Techniques

Graphing this function involves plotting points on a coordinate plane based on the transformation rules and understanding of domain and range.

To start:

• Identify several key \( x \) values that align with the function's domain. For \( y = \sqrt{x} - 1 \), good choices might be 0, 1, 4, and 9.
• Calculate the corresponding \( y \) values by plugging \( x \) into the function, which gives (0, -1), (1, 0), (4, 1), and (9, 2).
• On a coordinate grid, plot these points carefully.
• Draw a smooth curve that passes through these plotted points, beginning at (0, -1) and extending to the right and upwards.

By following these steps, you accurately reflect how the function behaves, and the graph clearly showcases the transformation effects.