Begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √(x+1)

Verified step by step guidance

1. Start by graphing the parent function f(x) = √x. This function is defined for x ≥ 0 and has a domain of [0, ∞) and a range of [0, ∞). The graph starts at the origin (0, 0) and increases slowly as x increases.
2. Understand the transformation applied to f(x) = √x to obtain g(x) = √(x + 1). The term (x + 1) inside the square root represents a horizontal shift. Specifically, adding 1 to x shifts the graph to the left by 1 unit.
3. To graph g(x) = √(x + 1), take each point on the graph of f(x) = √x and shift it 1 unit to the left. For example, the point (0, 0) on f(x) will move to (-1, 0) on g(x). Similarly, the point (1, 1) on f(x) will move to (0, 1) on g(x).
4. Verify the domain and range of g(x). Since the square root function is only defined for non-negative values inside the square root, the domain of g(x) is x ≥ -1. The range remains the same as the parent function, which is [0, ∞).
5. Sketch the graph of g(x) = √(x + 1) by plotting the transformed points and connecting them with a smooth curve. Ensure the graph starts at (-1, 0) and follows the same shape as the parent function, increasing slowly as x increases.

Key Concepts

Square Root Function

The square root function, f(x) = √x, is defined for x ≥ 0 and produces non-negative outputs. Its graph is a curve that starts at the origin (0,0) and increases gradually, reflecting the relationship between x and its square root. Understanding this function is crucial as it serves as the foundation for applying transformations.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For the function g(x) = √(x+1), the '+1' indicates a horizontal shift to the left by 1 unit. Recognizing how these transformations affect the original graph of f(x) is essential for accurately graphing the new function.

Domain and Range

The domain of a function refers to the set of all possible input values, while the range is the set of all possible output values. For the square root function, the domain is [0, ∞) and the range is also [0, ∞). When transforming the function to g(x) = √(x+1), the domain shifts to [-1, ∞) due to the horizontal shift, which is important for understanding the behavior of the graph.