What is the range of the function $y=\sqrt[3]{x+8}$ ?
The function given is \(y = \sqrt[3]{x+8}\). This is a cube root function. The cube root function \(f(u) = \sqrt[3]{u}\) is defined for all real numbers \(u \in (-\infty, \infty)\) because every real number has exactly one real cube root. In our case, the expression inside the root is \(u = x+8\). Since \(x+8\) is a linear expression defined for all real numbers, the entire function \(y = \sqrt[3]{x+8}\) is defined for all real numbers \(x\). Thus, the domain of the function is \((-\infty, \infty)\).
To determine the range of the function, we examine its behavior as the independent variable \(x\) varies. As \(x\) approaches positive infinity (\(x \to \infty\)), the term \(x+8\) also approaches infinity, and therefore its cube root \(\sqrt[3]{x+8}\) approaches positive infinity (\(y \to \infty\)). Similarly, as \(x\) approaches negative infinity (\(x \to -\infty\)), the term \(x+8\) approaches negative infinity, and its cube root \(\sqrt[3]{x+8}\) approaches negative infinity (\(y \to -\infty\)).
Because the cube root function is continuous everywhere on its domain, and it covers all values from negative infinity to positive infinity, it must take on every real number value. For any arbitrary real value \(y\), we can find a corresponding \(x\) by solving the equation \(y = \sqrt[3]{x+8}\). Cubing both sides, we get \(y^3 = x+8\), which implies \(x = y^3 - 8\). Since this formula gives a well-defined real number \(x\) for any real number \(y\), the range of the function is the set of all real numbers, denoted as \((-\infty, \infty)\).
Based on our analysis, the range of the function is all real numbers. Comparing this to the given options:
The correct option is \(-\infty < y < \infty\). Thus, the final answer is \(\boxed{-\infty < y < \infty}\).
In physics, cube root relationships often appear when dealing with volume-related measurements. For instance, if you are modeling the relationship between the volume \(V\) of a spherical object and its radius \(r\), the formula \(V = \frac{4}{3}\pi r^3\) can be inverted to find the radius if the volume is known: \(r = \sqrt[3]{\frac{3V}{4\pi}}\). Just as our function covers all real values in its output range, knowing the range of such functions helps scientists understand the possible dimensions of objects based on their properties, ensuring their models are mathematically consistent across all valid inputs.