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The domain of a function tells us which values we can plug into the function that will not cause any mathematical errors. For our function, \( f(x) = -1 + \sqrt{x+2} \), we must be cautious about the square root. The expression inside the square root, \( x+2 \), must be greater than or equal to zero to avoid taking the square root of a negative number, which is undefined in the real number system.
We set up the inequality: \( x+2 \geq 0 \). Solving this inequality involves subtracting 2 from both sides, resulting in \( x \geq -2 \).
This means that the domain of this function includes all real numbers from \( -2 \) to positive infinity. We write this domain in interval notation as \([-2, \infty)\). Here, \(-2\) is included in the domain because the expression is zero at \( x = -2 \), and the square root of zero is zero.

The range of a function is about the possible output values that the function can produce. For \( f(x) = -1 + \sqrt{x+2} \), we start by considering the smallest and largest possible values that can result from the expression \( \sqrt{x+2} \).
At \( x = -2 \), the expression equals zero, so \( f(-2) = -1 + 0 = -1 \). This value represents the minimum output, or the smallest the function can be.

• As \( x \) increases, the value of \( \sqrt{x+2} \) becomes larger, potentially reaching very large numbers. Since there is no upper bound for \( x \), the outputs of the function can increase without limit.

Therefore, the range of the function stretches from \(-1\) upward to infinity, noted in interval notation as \([-1, \infty)\).

Zeros of a function are the input values where the function outputs zero. These are also referred to as the "roots" or "x-intercepts."

To find the zeros of \( f(x) = -1 + \sqrt{x+2} \), we set the function equal to zero: \(-1 + \sqrt{x+2} = 0\).

• Adding 1 to both sides gives us: \( \sqrt{x+2} = 1 \).
• Squaring both sides removes the square root: \( x+2 = 1^2 = 1 \).
• Finally, subtracting 2 from both sides results in \( x = -1 \).

So, \( f(x) = 0 \) when \( x = -1 \), meaning the zero of the function is at that point. This tells us that the graph of the function will cross the x-axis at \( x = -1 \).

Square root functions have unique properties because they involve the square root of an expression. These functions have a distinctive curved graph and specific domain restrictions due to the square root.

For example, in the function \( f(x) = -1 + \sqrt{x+2} \), the square root affects both the domain and the shape of the graph.

• The domain is limited to x-values that make the quantity inside the square root non-negative. Thus, any changes inside the radical affect the domain directly.
• The graph of the function starts at a certain point (here, \( x=-2 \)) and continues to rise that impacts the function's minimum or starting value (here, \(-1\)).

Understanding these features helps in graphing and solving equations involving square root functions. These functions typically resemble one half of a parabola, reflecting only the positive arc due to the square root's nature.