Problem 42 - Algebra 2

Question

Graph. Find the domain and the range of each function. \(y=7-\sqrt{2 x-1}\)

Step-by-Step Solution

Answer

The domain of the function \(y=7-\sqrt{2x-1}\) is \([0.5, \infty)\), and the range is \((-\infty, 7]\)

Step 1: Determine the Domain

To find the domain, set the expression under the square root greater than or equal to zero. Solve the inequality \(2x-1 \geq 0\) for \(x\) to get \(x \geq 0.5\).

Step 2: Determine the Range

In the given function \(y=7-\sqrt{2x-1}\), the original square root function is shifted downward by 7 units, and it opens downwards because of the negative sign before the square root. Hence, for such a function, the largest value of \(y\) will be the y-coordinate of the vertex of the graph, which is 7 here. Thus, the range of the function is \(-\infty, 7\], i.e., \(y\) is less than or equal to 7.

Step 3: Graphing the Function

First, plot the point (0.5, 7), which is the vertex of the graph. Since the function opens downwards, the graph will decrease. Remember that the square root graph is parabolic. After plotting this on a graph, the solutions obtained in the previous steps will be evident.

Key Concepts

Graphing Functions

Graphing functions is a fundamental skill that helps us visually understand how a function behaves. When graphing a function, you're looking to plot its values on a coordinate system. For our function, \(y=7-\sqrt{2x-1}\), graphing it allows us to clearly see its domain and range.

The key steps in graphing any function include:

• Determining key points: Start by finding specific points on the graph, like vertices or intercepts, that can help build the rest of the graph.
• Understanding transformations: Recognize any shifts in the graph due to modifications like adding or subtracting constants.
• Connecting the points: Use the known points and transformations to sketch the curve of the graph.

For the provided function, plot the initial point (0.5, 7). This is crucial because it's the highest point in this case. As you plot the graph, keep in mind this function's square root part will have an influence, making the graph decrease further as \(x\) increases.

Inequalities

Inequalities play an essential role when identifying the domain of functions that have square root symbols. Why is this? Simply put, when dealing with square roots, we must ensure that the value under the square root is non-negative.

For instance, with \(y=7-\sqrt{2x-1}\), we need to solve the inequality \(2x-1 \geq 0\) to find valid values for \(x\) such that the square root is well-defined. The process is straightforward:

• Start with the inequality given by the square root condition: \(2x-1 \geq 0\).
• Solve it: Add 1 to both sides to get \(2x \geq 1\), then divide both sides by 2, leading to \(x \geq 0.5\).

This inequality solution outlines the domain of the function, indicating that \(x\) must be 0.5 or greater.

Square Root Functions

Square root functions have unique characteristics. In this case, the function involves a negative square root which impacts how the function is graphed and interpreted.

A square root function typically looks like \(y=\sqrt{x}\), and its graph appears as a curve starting at the vertex and gradually increasing. However, our function \(y=7-\sqrt{2x-1}\) presents some differences:

• The negative sign before the square root suggests the graph opens downward, inverting the typical "/" shape.
• The function is also vertically translated upward by 7 units, indicating the starting point (vertex) is at the height of 7 for \(x=0.5\).

The range, as a result, is impacted by these transformations. Because the highest point is the vertex at \(y=7\), the function decreases. Therefore, the range is set from \(-\infty\) up to and including 7. Understanding these square root function properties enables us to better interpret and graph functions like this with accuracy.