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The square root function is a fundamental concept in mathematics. Its standard form is represented as \( y = \sqrt{x} \). This function is notably characterized by its starting point at the origin \((0, 0)\). It only yields real values when \( x \geq 0 \), meaning our inputs (or, domain) must be non-negative numbers.

Graphically, the square root function has a unique shape. It begins at the origin and curves upward, displaying a rapid increase at first, which then slows as \( x \) grows larger. This creates a half-parabola curve that opens to the right.

In its basic form, the range is also constrained. For the basic \( y = \sqrt{x} \) function, the outputs are all non-negative, so the range is \([0, \infty)\).

The square root function is particularly important because it introduces concepts like the domain and range restrictions, and is the basis for understanding more complex functions like transcendental or exponential functions.

Function transformation allows us to take a basic function and adjust its graph for various parameters. For the function \( y = 10 \sqrt{x-20} + 5 \), several transformations apply to the base square root function \( y = \sqrt{x} \).

• Horizontal Shift: The transformation \( \sqrt{x-20} \) indicates a horizontal shift 20 units to the right. This is because the expression inside the square root changes the starting point of the graph.
• Vertical Stretch: The multiplication by 10 causes a vertical stretch of the graph, making it rise faster.
• Vertical Shift: Adding 5 outside the square root shifts the graph upwards by 5 units.

The endpoint for these transformations can be determined by solving the equation inside the square root for zero, \( x - 20 = 0 \). Hence, the transformed graph starts at the point \((20, 5)\) and then follows the characteristic shape of a square root function, but now influenced by these shifts and stretches.

Understanding the domain and range of a function is crucial, as it defines the valid inputs and possible outputs. For the function \( y = 10 \sqrt{x-20} + 5 \), the domain is determined by the expression inside the square root. Since square roots are only defined for non-negative values, \( x - 20 \geq 0 \), hence the domain is \( x \geq 20 \). In interval notation, this is indicated as \([20, \infty)\).

The range is tackled by considering how the transformations affect potential outputs. The minimum value of \( y \) occurs at its endpoint. When \( x = 20 \), \( y = 10 \times \sqrt{0} + 5 = 5 \). Thus, as \( x \) increases, the function outputs grow because the square root increases and is scaled by the factor of 10, pushing up the resultant \( y \)-values. Hence, the range for the function is \([5, \infty)\).

Both the domain and range reflect how the transformed graph behaves, guiding us in predicting the function's behavior for different inputs.