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The domain of a function consists of all the possible input values, or 'x' values, which will not result in errors during function calculations. Understanding the domain is crucial because it tells you where the function is defined and where it isn't.
For the function \( g(x) = \sqrt{x+4} \), the expression inside the square root must be non-negative, as the square root of a negative number is not defined in the set of real numbers.
To find the domain, solve the inequality:

• \( x + 4 \geq 0 \)
• This simplifies to \( x \geq -4 \)

Thus, the domain of this function is \([ -4, \infty)\), meaning any \( x \) value greater than or equal to -4 is valid.
Finding the domain ensures that you use appropriate x-values when evaluating the function with no errors or undefined results.

A square root function typically takes the form \( f(x) = \sqrt{g(x)} \), where \( g(x) \) must be a non-negative expression.
This is because you cannot find the real square root of a negative number. When graphing square root functions, you often observe a parabola-like curve, but only one "leg" of it.
For the function \( g(x) = \sqrt{x+4} \), the square root function starts at the point where the expression inside the square root is zero. Here, it begins at \( x = -4 \) because that’s when \( x + 4 = 0 \).
The graph will then stretch to the right and upwards as \( x \) increases and \( g(x) \) values become larger.
Key Characteristics:

• Starts at a specific \( x \)-value, here \( x = -4 \).
• Continues indefinitely to the right.
• Increases gradually.

Understanding the shape and starting point of a square root function graph helps determine the behavior of the function.

The coordinate plane is a vital tool in graphing functions. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis) intersecting at the origin (0,0).
This two-dimensional surface allows us to visually represent mathematical functions by plotting points that correspond to function values.
When plotting the graph of \( g(x) = \sqrt{x+4} \), use the coordinate plane to place each calculated point derived from the function:

• Start by identifying the domain, as we did with \([ -4, \infty) \).
• Select points within this domain to evaluate, like \( x = -4, -3, 0, 1, 5 \).
• Calculate the corresponding \( g(x) \) values to obtain points such as \( (-4, 0), (-3, 1), (0, 2), (1, \sqrt{5}), (5, 3) \).

Then, plot these points on the coordinate plane and connect them smoothly.
This gives the visual outline of the function, providing insight into the behavior and direction of the graph.
Utilizing the coordinate plane effectively allows for clearer understanding and interpretation of mathematical relationships present in functions.