91Ó°ÊÓ

A square root function is defined as a function that includes the square root of a variable. For a function like \( g(x) = 4 \sqrt{x} \), it's important to understand its domain—the set of all possible values of \( x \) that can be input into the function without resulting in an undefined or imaginary number.

Since we cannot take the square root of a negative number in real number mathematics, the domain of a square root function must include only non-negative values of \( x \). Hence, for the function \( g(x) = 4 \sqrt{x} \), the domain is:

• \( x \geq 0 \)

This means that \( x \) can be zero or any positive number. To find the domain, simply set the expression inside the square root greater than or equal to zero and solve for \( x \). For example:

• \( x \geq 0 \).

Therefore, when you're plotting the graph, you'll only include values of \( x \) that are zero or positive.

To make a graph, identifying and plotting key points can make the overall shape clearer and simplify the graphing process. For the function \( g(x) = 4 \sqrt{x} \), it's helpful to calculate \( g(x) \) at specific values of \( x \).

Consider values of \( x \) that are perfect squares because they simplify the calculations:

• At \( x = 0 \), \( g(0) = 4 \sqrt{0} = 0 \).
• At \( x = 1 \), \( g(1) = 4 \sqrt{1} = 4 \).
• At \( x = 4 \), \( g(4) = 4 \sqrt{4} = 8 \).
• At \( x = 9 \), \( g(9) = 4 \sqrt{9} = 12 \).

By plotting these points: \( (0, 0) \), \( (1, 4) \), \( (4, 8) \), and \( (9, 12) \) on a coordinate plane, you build a visual framework of the function's graph. These points provide a snapshot of the curve's behavior and make it easier to draw the final graph accurately.

Square root functions have unique characteristics that differentiate their graphs from other types of functions. The general form of a square root function is \( g(x) = a \sqrt{x} \). Here are the key characteristics:

• Starting Point: The graph starts at the origin \( (0, 0) \) for \( g(x) = a \sqrt{x} \).
• Direction: Since the function values are derived from taking the square root of \( x \), the graph moves upwards to the right from the origin.
• Shape: The graph has a curve that becomes less steep as \( x \) increases. Initially steep, flattening as it extends further along the x-axis.
• Range: The possible output values \( g(x) \) begin from 0 and extend upwards, i.e., \( g(x) \geq 0 \).
• Positive Y-values: The graph will only have non-negative y-values because the square root and the coefficient \( a \) ensure all outputs are non-negative.

These characteristics help identify and differentiate the graph of a square root function from others. By understanding these properties, you can better visualize and sketch the graph, ensuring it correctly represents the underlying mathematics.