Graph each function. See Examples 6 – 8. _ ƒ(x) = 2√x + 1

Verified step by step guidance

1. Identify the basic function: The function \( f(x) = 2\sqrt{x} + 1 \) is a transformation of the square root function \( \sqrt{x} \).
2. Determine the transformations: The function \( 2\sqrt{x} + 1 \) involves a vertical stretch by a factor of 2 and a vertical shift upwards by 1 unit.
3. Find the domain: Since the square root function is only defined for non-negative values, the domain of \( f(x) \) is \( x \geq 0 \).
4. Plot key points: Calculate a few key points by substituting values of \( x \) into the function, such as \( x = 0, 1, 4 \), to find corresponding \( f(x) \) values.
5. Sketch the graph: Use the key points and transformations to sketch the graph, starting from the origin and following the shape of the square root function, stretched and shifted as determined.

Key Concepts

Square Root Function

The square root function, denoted as √x, is defined for non-negative values of x and produces the principal square root. It is a fundamental function in mathematics, characterized by its gradual increase and a domain of [0, ∞). Understanding its properties, such as its shape and behavior, is essential for graphing functions that involve square roots.

Transformation of Functions

Transformations of functions involve shifting, stretching, or compressing the graph of a function. In the given function ƒ(x) = 2√x + 1, the '2' indicates a vertical stretch by a factor of 2, while the '+1' represents a vertical shift upward by 1 unit. Recognizing these transformations helps in accurately graphing the function based on its parent function.

Domain and Range

The domain of a function refers to the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). For the function ƒ(x) = 2√x + 1, the domain is [0, ∞) since square roots are only defined for non-negative numbers, and the range is [1, ∞) due to the vertical shift. Understanding domain and range is crucial for graphing and interpreting functions.