Problem 12 - Algebra 1: Concepts and Skills

Question

Find the domain and the range of the function. $$y=\sqrt{x}+6$$

Step-by-Step Solution

Answer

Domain: \(x \geq 0\), Range: \(y \geq 6\)

Step 1 Determine the domain

The domain of a function is the set of all possible input values (often 'x' values) which will produce a valid output from a particular function. In this case, because we are dealing with the square root function, and since you cannot take the square root of a negative number within the real number system, the domain is all x for which \(x\geq 0\). Therefore, the domain is \(x \geq 0\).

Step 2 Determine the range

The range of a function is the complete set of all possible resulting values of the dependent variable (often 'y' or 'f(x)' values), after we have substituted the domain. In our function \(y=\sqrt{x}+6\), the smallest value is attained when x = 0, so y = 6. Because the square root function outputs only non-negative values, the rest of possible y-values are greater than 6. Therefore, the range of the function is \(y \geq 6\).