In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. h(x) = √(x+1)-1

Verified step by step guidance

1. Start by graphing the basic square root function, $f(x)=\sqrt{x}$. This graph starts at the origin (0,0) and increases slowly, forming a curve that moves to the right.
2. Identify the transformations needed to graph $h(x)=\sqrt{x+1}-1$. Notice that the function inside the square root, $x+1$, indicates a horizontal shift.
3. The term $x+1$ inside the square root suggests a horizontal shift to the left by 1 unit. This means every point on the graph of $\sqrt{x}$ will move 1 unit to the left.
4. The $-1$ outside the square root indicates a vertical shift. This means the entire graph will move down by 1 unit.
5. Combine these transformations: shift the graph of $\sqrt{x}$ left by 1 unit and down by 1 unit to obtain the graph of $h(x)=\sqrt{x+1}-1$.

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 in the given problem.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting the graph of a function. In this case, the function h(x) = √(x+1) - 1 represents a horizontal shift to the left by 1 unit and a vertical shift downward by 1 unit. Mastery of these transformations allows for the accurate graphing of new functions based on known ones.

Function Composition

Function composition refers to the process of applying one function to the results of another. In the context of h(x), the inner function (x + 1) modifies the input of the square root function, while the outer function (subtracting 1) adjusts the output. Understanding how to compose functions is essential for manipulating and graphing complex functions derived from simpler ones.