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The domain and range of a function are crucial in understanding the behavior of its input and output values. The **domain** refers to the set of all possible input values (usually 'x') for which the function is defined. For the function \( f(x) = \sqrt{x+1} \), in order for the square root to be real, the expression inside the square root must be non-negative. Thus, we have the condition \( x + 1 \geq 0 \) or \( x \geq -1 \). Therefore, the domain of \( f(x) \) is \([-1, \infty)\).

The **range** refers to the set of all possible output values (usually 'f(x)') the function can produce. Since a square root function only generates non-negative results, the smallest value \( f(x) \) can attain is 0, which happens when \( x = -1 \). As \( x \) increases, so does \( f(x) \). Consequently, the function's range is \([0, \infty)\).

When we switch to the inverse function \( f^{-1}(x) = x^2 - 1 \), the role of domain and range flips. The domain of \( f^{-1} \) is \([0, \infty)\) while the range becomes \([-1, \infty)\). This switch happens because the inverse function essentially reverses the input and output roles in the original function.

Graph sketching is a vital technique for visualizing a function's behavior. It allows us to see how the function behaves across its domain and range. To sketch the graph of \( f(x) = \sqrt{x+1} \), start by locating its starting point. Since \( x \geq -1 \), substituting \( x = -1 \) gives the point \((-1, 0)\).

From this point, observe the general tendency of a square root function, which increases and forms a curve that bends downward, indicating a concave shape. This is because while the function is increasing, the rate of increase slows down as \( x \) grows larger, creating a slow, smooth rise.

To sketch \( f^{-1}(x) \), reflect the graph of \( f(x) \) across the line \( y = x \). This line acts as a mirror, and each point on \( f(x) \) translates over that line. The original point \( (-1, 0) \) becomes \( (0, -1) \), and this reflection characterizes how an inverse function behaves graphically.

The square root function is a specific type of function that commonly appears in mathematics. Its general form is \( f(x) = \sqrt{x} \). For \( f(x) = \sqrt{x+1} \), it involves a horizontal shift of the basic square root graph.

A shift to the left by 1 unit is performed, resulting from the \( +1 \) inside the square root. Hence, the graph starts at \( x = -1 \) rather than \( x = 0 \). This modification affects both the domain and how the graph is shaped, compared to a simple square root curve starting at the origin.

Moreover, the square root function typically portrays a curve that increases but is asymptotic to both axes. As the function continues, it grows slowly due to the flattening nature of the curve as \( x \) approaches infinity. Understanding these characteristics helps anticipate how alterations to the function, such as shifts, influence its overall form and properties.