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The domain of a function is a fundamental concept in mathematics that refers to all the possible input values (usually represented by \( x \)) for which the function is defined. In simpler terms, it’s the set of all \( x \) values that you can plug into the function without breaking any mathematical rules. For the function \( f(x) = \sqrt{x+2} - 3 \), we need to determine where the expression under the square root is non-negative because square roots of negative numbers are not real. Since

• the expression inside the square root is \( x+2 \),
• we require \( x + 2 \geq 0 \).

This inequality simplifies to \( x \geq -2 \). Therefore, the domain of our function is \([-2, \infty)\). This means you can use any \( x \) value starting from -2 and going upwards without limit.
Understanding the domain helps in predicting where the graph of the function will exist on the \( x \)-axis.

Transformations of functions involve shifting or stretching the graph of the original function to create a new graph. These adjustments change the appearance of the function on the graph without altering its basic shape. For the function \( f(x) = \sqrt{x+2} - 3 \), we recognize this function as a transformation of the parent function \( \sqrt{x} \). Here, the transformations include:

• Horizontal Shift: The \( x \) inside the square root is modified as \( x+2 \). This indicates a shift 2 units to the left because adding a number inside the function translates it horizontally in the opposite direction.
• Vertical Shift: The whole function is reduced by 3, represented by \( -3 \) outside the square root. This means the graph moves 3 units downwards.

When you graph it, these combined shifts result in a curve that starts at a new position compared to the basic square root function. The graph retains the square root's curve but appears moved both horizontally and vertically.

The range of a function refers to all the possible output values (usually represented by \( y \)) a function can produce. It tells us how low or high the function can go on the \( y \)-axis. For \( f(x) = \sqrt{x+2} - 3 \), knowing the transformations it underwent helps in determining its range.

• The smallest value \( f(x) \) can have is determined when \( x = -2 \), leading to \( f(-2) = \sqrt{0} - 3 = -3 \).
• As \( x \) becomes larger (greater than -2), \( \sqrt{x+2} \) increases, pushing \( f(x) \) upward indefinitely.

Thus, the range of this function is \([-3, \infty)\). This means the lowest that \( f(x) \) can go is -3, but it can get arbitrarily large as \( x \) increases. The concept of range is critical when graphing, as it defines where the graph will lie on the horizontal dimension.