Problem 35 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. Then use severalvalues in the domain to make a table of values for the function. $$y=\sqrt{x+9}$$

Step-by-Step Solution

Answer

The domain of the function \(y=\sqrt{x+9}\) is \([-9, +\infty)\). The table of values of the function includes {(-9,0), (-4, \sqrt{5}), (0, \sqrt{9}), (1, \sqrt{10}), (10, \sqrt{19})}.

Step 1: Finding the Domain

The expression under the square root must be greater than or equal to zero. This gives the inequality \(x+9 \geq 0\). Solving this would give us the domain of the function. Solving for \(x\) gives \(x \geq -9\). Hence the domain of the function is \([-9, +\infty)\).

Step 2: Tabulating the Function Values

Choose several values from the domain and substitute them into the equation. For instance, put \(x = -9\), then \(y = \sqrt{-9+9} = 0\). Continuing this way, with \(x= -4, 0, 1, 10\), we can form pairs of \(x\) and calculate \(y\) respectively.

Step 3: Table of Values

After substituting the selected values of \(x\), the table of values will look something like this: {(-9,0), (-4, \sqrt{5}), (0, \sqrt{9}), (1, \sqrt{10}), (10, \sqrt{19})}. The selected values of \(x\) and the corresponding \(y\) values form an ordered pair that can be graphed on a coordinate plane.

Key Concepts

Square Root Function

The square root function is a special type of function in mathematics known for its unique properties. In this specific case, the function is given by the expression \(y = \sqrt{x + 9}\). This function indicates the principal square root of \(x + 9\), meaning it’s only valid for non-negative values of \(x + 9\). Therefore, any value of \(x\) substituted into the function must result in a non-negative number under the square root.
To find the domain of this function, we solve the inequality \(x + 9 \geq 0\), which means \(x\) must be greater than or equal to \(-9\). Hence, the domain of the square root function here is \([-9, +\infty)\).
This domain points out that for any input \(x\) within this range, a real-valued output \(y\) will exist. This characteristic of the square root function is vital in defining which \(x\) values are permissible to use.

Function Values Table

After determining the domain of a function, preparing a table of values is the next logical step. This involves selecting various \(x\) values from the domain and calculating their corresponding \(y\) values using the given function. For our function \(y = \sqrt{x + 9}\), this means inputting values of \(x\) that are greater than or equal to \(-9\).

• Begin by choosing some specific values for \(x\): for example, \(-9\), \(-4\), \(0\), \(1\), and \(10\).
• Substitute these \(x\) values into the function to find the associated \(y\) values.

For instance, at \(x = -9\), the computation yields \(y = \sqrt{-9 + 9} = 0\). Similarly, at \(x = -4\), \(y = \sqrt{-4 + 9} = \sqrt{5}\). Continuing this way allows you to compile these \((x, y)\) pairs into a clear, structured representation. This table of values serves as a foundational tool for visually plotting the function's behavior.

Coordinate Plane

Using ordered pairs from the function values table on a coordinate plane is a powerful method for visualizing the function's graph. The coordinate plane consists of two axes, the horizontal axis (x-axis) which represents \(x\), and the vertical axis (y-axis) for \(y\).
When plotting, each ordered pair \((x, y)\) corresponds to a point on this grid:

• The \(x\)-coordinate shows how far along the x-axis the point is.
• The \(y\)-coordinate represents its distance along the y-axis.

For \(y = \sqrt{x + 9}\), plotting takes our calculated pairs like \((-9, 0)\), \((-4, \sqrt{5})\), \((0, 3)\), \((1, \sqrt{10})\), \((10, \sqrt{19})\) and places them on the plane.
Connecting these points with a smooth curve will illustrate how the function behaves as \(x\) changes. The function's graph will reflect its domain properties and range, starting from the point \(-9\) on the x-axis and extending infinitely to the right, showcasing the non-negative values dictated by the function's structure.