Question 2 50 pts Identify the domain and range of the following radical function. \[ y=\sqrt[3]{x}-3 \] Domain: \( x \geq 0 \) Range: \( y \geq 1 \) Domain: \( x \geq 1 \) Range: \( y \geq 3 \) Domain: All Real Numbers Range: All Real Numbers Domain: \( x \geq 3 \) Range: \( y \geq-1 \)
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The function \(y = \sqrt[3]{x} - 3\) is a cubic root function shifted downward by 3 units. Unlike square root functions, cubic root functions have a domain that includes all real numbers, since you can take the cube root of any real number without restriction. The range of this function is also all real numbers. As \(x\) approaches negative infinity, \(y\) will approach negative infinity as well, and as \(x\) approaches positive infinity, \(y\) will approach positive infinity. So, you can put on your math party hats—this function is open for business across the whole real number line! If you're ever in doubt about domain and range, remember that the roots of odd degrees (like cube roots) are friendly characters, inviting every real number to join the party!
