Problem 70 - Algebra 2

Question

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

Step-by-Step Solution

Answer

Domain: \([2, \infty)\); Range: \([0, \infty)\). Graph starts at (2,0), curves upward to the right.

Step 1: Identify the Function Type

The given function is a square root function, which is represented as \(y=\sqrt{x-2}\). Such functions often involve determining the domain based on when the expression under the square root is non-negative.

Step 2: Determine the Domain

To find the domain of \(y=\sqrt{x-2}\), set the expression inside the square root greater than or equal to zero: \(x-2 \geq 0\). Solving for \(x\) gives \(x \geq 2\). Thus, the domain is \([2, \infty)\).

Step 3: Determine the Range

Since the square root function produces non-negative outputs, and the minimum value of \(y\) occurs when \(x = 2\), \(y = \sqrt{2-2} = 0\), the range of \(y=\sqrt{x-2}\) is \([0, \infty)\).

Step 4: Graph the Function

Start by plotting the point where the function begins at \(x=2\) and \(y=0\). Then plot additional points for a few values of \(x > 2\). For instance, \(x=3\) gives \(y=\sqrt{3-2}=1\) and \(x=6\) gives \(y=\sqrt{6-2}=2\). Connect the points with a smooth curve extending to the right.

Key Concepts

Domain

The domain of a function describes all the possible values that the independent variable, typically represented by \(x\), can take. For the square root function \(y = \sqrt{x-2}\), we must ensure that the expression under the square root is non-negative, because you cannot take the square root of a negative number in the set of real numbers. Therefore:

• The expression \(x-2\) must satisfy \(x-2 \geq 0\).
• Solve the inequality: \(x \geq 2\).

This means our domain includes all real numbers starting from 2 and extending to infinity, which we write as the interval \([2, \infty)\).
Always check the conditions of the problem, especially with square roots, to properly determine the domain.

Range

The range of a function is the set of all possible output values, typically represented by \(y\). For the function \(y = \sqrt{x-2}\), the range is determined by the possible values that \(y\) can take. This square root function only gives non-negative outputs:

• When \(x = 2\), \(y = \sqrt{2-2} = 0\); hence the function starts at 0.
• As \(x\) increases, \(y\) becomes larger, with no upper limit.

Consequently, the range is represented by the interval \([0, \infty)\), which means \(y\) can be any non-negative real number. Understanding the range involves knowing the behavior of the function's rule, especially where expressions like square roots are involved.

Graphing Functions

Graphing a function helps visualize its behavior. For \(y = \sqrt{x-2}\), the function begins at the point \((2, 0)\) and increases gradually.

• Start by plotting the initial point where \(x = 2\) and \(y = 0\). This is where the function "begins."
• Select a few more values of \(x\) to find corresponding \(y\) values; for example, \(x = 3\) gives \(y = 1\) and \(x = 6\) produces \(y = 2\).

Plot these points on a Cartesian plane. The graph will show a gentle curve extending to the right from \((2, 0)\), reflecting the incremental increase typical of square root functions. This visual representation shows that as \(x\) increases, \(y\) increases slowly, with the curve hitting neither a maximum nor reaching negative \(y\) values.

Expression Under the Square Root

The expression under the square root, or the "radicand," fundamentally affects the properties of the square root function, organizing the domain and path of the graph.

• For the function \(y = \sqrt{x-2}\), the expression \(x-2\) must be non-negative.
• This condition, \(x-2 \geq 0\), directly informs the domain of the function, ensuring that values computed by the square root are real and meaningful.

When analyzing such functions, always begin with setting the radicand greater than or equal to zero to ensure the function maintains valid outputs in the real number system. This analysis underpins entire functional behavior and must not be overlooked.