Given the function, answer the questions.
[ h(x) = \sqrt{x^3-27} ]
- What is the domain of \(h(x)\)?
- What is the range of \(h(x)\)?
- Find \(h^{-1}(x)\)?
- What is the domain and range of \(h^{-1}(x)\)?
Solution
- \([3,\infty)\)
- \([0,\infty)\)
3. \[ \begin{{array}}{{rcl}} y&=&\sqrt{x^3-27}\\ \text{Swap }x\text{ and }y\\ x&=&\sqrt{y^3-27}\\ x^2&=&y^3 - 27\\ x^2+27 &=&y^3\\ \sqrt[3]{x^2+27} &=&y\\ h^{-1}(x) &=&\sqrt[3]{x^2+27} \end{{array}} \]
4. Domain: \([0,\infty)\) Range:\([3,\infty)\)