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)

Verified step by step guidance

1. Start by graphing the parent function \( f(x) = \sqrt{x} \). This is the basic square root function, which starts at the origin (0, 0) and increases gradually as x increases. The domain of this function is \( x \geq 0 \), and the range is \( y \geq 0 \).
2. Next, analyze the given function \( h(x) = \sqrt{-x + 1} \). Rewrite it as \( h(x) = \sqrt{-(x - 1)} \) to make the transformation clearer. This shows that the function involves a horizontal shift and a reflection.
3. The term \( -(x - 1) \) indicates two transformations: (1) a horizontal reflection across the y-axis due to the negative sign in front of \( x \), and (2) a horizontal shift to the right by 1 unit because of \( x - 1 \).
4. Apply these transformations to the graph of \( f(x) = \sqrt{x} \): (1) Reflect the graph of \( \sqrt{x} \) across the y-axis, and (2) shift the resulting graph 1 unit to the right. This gives the graph of \( h(x) = \sqrt{-(x - 1)} \).
5. Finally, determine the domain and range of \( h(x) \). The domain is restricted by the square root: \( -(x - 1) \geq 0 \), which simplifies to \( x \leq 1 \). The range remains \( y \geq 0 \), as square root values are always non-negative.

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, reflecting, stretching, or compressing the graph of a function. For the function h(x) = √(-x + 1), the transformations include a horizontal reflection across the y-axis and a horizontal shift to the right by 1 unit. Mastery of these transformations allows for the accurate graphing of modified functions based on their parent functions.

Function Composition

Function composition refers to the process of applying one function to the results of another. In the context of transformations, understanding how to compose functions helps in visualizing how changes to the input (x-values) affect the output (y-values). This concept is essential for accurately interpreting and graphing the transformed function h(x) based on the original square root function.