Problem 12 - Algebra 2

Question

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

Step-by-Step Solution

Answer

Domain: \([0, \infty)\), Range: \([0, \infty)\).

Step 1: Identify the Function Type

The given function is a square root function in the form of \[y = \frac{1}{2} \sqrt{x}\] which is a transformation of the basic square root function, \(y = \sqrt{x}\). Square root functions are known to have specific characteristics and are generally defined for non-negative values of \(x\) due to the square root.

Step 2: Determine the Key Features

Identify the domain of the function by considering the values of \(x\) for which the function is defined. Since we cannot take square roots of negative numbers in the real number system, \(x \geq 0\). For the range, considering the output of the square root function is always non-negative, and it is scaled by \(\frac{1}{2}\), the range is \(y \geq 0\).

Step 3: Create a Table of Values

Choose values for \(x\), and calculate the corresponding \(y\) values to help plot the graph.For example:- If \(x = 0\), \(y = \frac{1}{2}\sqrt{0} = 0\)- If \(x = 1\), \(y = \frac{1}{2}\sqrt{1} = \frac{1}{2}\)- If \(x = 4\), \(y = \frac{1}{2}\sqrt{4} = 1\)These points will allow us to accurately sketch the graph.

Step 4: Plot the Points and Sketch the Graph

Using the points from the table of values, plot them on a coordinate plane. Connect these points smoothly starting from the origin and curving upwards to the right.

Step 5: State the Domain and Range

From the information gathered in Step 2 and the sketched graph:- Domain: The function is defined for all non-negative \(x\) values, so the domain is \([0, \infty)\).- Range: Since the output \(y\) starts from 0 and increases, the range is \([0, \infty)\).

Key Concepts

Domain and Range

Understanding the domain and range of a function helps you know where the function lives on the graph. The **domain** refers to all the possible input values (x-values) that the function can accept, while the **range** is all the possible output values (y-values) that the function can produce.
In the function \(y = \frac{1}{2}\sqrt{x}\), which is a type of square root function, the domain is determined by the restriction that you cannot take the square root of negative numbers. This means the function is only defined for non-negative values of \(x\), or in simpler terms: **\(x \geq 0\)**.
Thus, the domain is \([0, \infty)\).
For the **range**, you look at what values \(y\) can take. Since the square root function always results in a non-negative number and it's further multiplied by \(\frac{1}{2}\), the smallest value \(y\) can be is zero. As \(x\) increases, \(y\) will also increase but will never become negative. Hence, the range is also \([0, \infty)\).

Square Root Function

The square root function is fundamental in algebra and precalculus. It generally appears in the form \(y = \sqrt{x}\). This type of function has a distinct shape and properties.

• It begins at the origin, point (0,0), because the square root of zero is zero.
• It travels in a smooth, gentle curve upwards and to the right, indicating an increase in \(y\) as \(x\) becomes larger.
• It is only defined for non-negative x-values, which confines the entire graph to the right of the y-axis.
• The value of \(y\) never becomes negative as you are taking the square root of x.

The square root function reflects certain real-world scenarios where negative outputs aren't possible.
For example, calculating length or distance always provides non-negative values. When graphing \(y = \sqrt{x}\), expect this typical curve that never dips below the x-axis.

Function Transformation

Function transformations involve altering the parent function to produce a new one. In **\(y=\frac{1}{2}\sqrt{x}\)**, a transformation of the basic square root function occurs.

• **Vertical Scaling:** The multiplication by \(\frac{1}{2}\) scales the graph vertically, making it "slower" to rise compared to \(y = \sqrt{x}\). This means for the same x-values, the outputs (y-values) are half as large.
• **No Horizontal Shift or Reflection:** Since there are no additional terms added inside the square root or outside, there's no horizontal shift or reflection across the axes.

Visualizing allows you to see that as \(\frac{1}{2}\) can be considered a compression of the function towards the x-axis or making the output half of what it would be based on the original square root function. By understanding this, you respect how the function behaves differently while maintaining its essential characteristics of a standard square root function.