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The range of a function is the complete set of all possible resulting values of the dependent variable (in this case, 'y'), after having substituted the domain (or the independent variable 'x') into the function. For the square root function given by the formula \( y = a \sqrt{x} \), the range is \( y \geq 0 \) because the square root of a number is defined only for non-negative numbers and results in non-negative outputs. This is a fundamental property of square roots that restricts the output of the function.

When examining any function, particularly square root functions, identifying the range is a crucial step in understanding the behavior of the function. Since square root functions involve only non-negative outputs, the lowest value of \( y \) when \( a \) is positive is zero, which occurs at \( x = 0 \). The range then extends to infinity as \( x \) increases because the square root of larger numbers gets larger.

In the context of the function \( y = a \sqrt{x} \), the term 'a' is referred to as the proportionality constant. A proportionality constant is a multiplier that scales the output of the function in direct correspondence to the input. It is a fixed number that determines how the dependent variable changes with respect to the independent variable.

In simpler terms, if 'a' is increased, the value of \( y \) will be larger for a given \( x \) compared to if 'a' is smaller. The sign of 'a' also affects the direction of the scaling; if 'a' is positive, it fits with the original function's range of non-negative values. Should 'a' be negative, it would flip the graph over the x-axis, resulting in negative outputs for the function. However, since the range of the function is given as \( y \geq 0 \), 'a' must be non-negative in this context.

Non-negative numbers are a set of numbers that include all positive numbers and zero. They are the result of a square root function where the domain (input values) is also non-negative. This is an important concept because square roots are undefined for negative numbers within the real number system.

Any square root function like \( y = a \sqrt{x} \) will only yield non-negative results because a square root extricates the original value that was squared to produce the input number 'x.' Since the operation of squaring a number (multiplying it by itself) always results in a non-negative product, taking the square root of that product yields a non-negative number. Hence when \( x \) is less than zero, the square root is not considered because it would result in imaginary numbers, which are beyond the scope of this function's definition.