For the function \(f(x) = x^{1/6} = \sqrt[6]{x}\), we note that it involves an even root. An even root is only defined for non-negative real numbers because the power of an even number is always non-negative. Therefore, the domain of \(f(x)\) is \(x \geq 0\), or in interval notation, \([0, \infty)\). As \(x\) increases from 0 to \(\infty\), \(\sqrt[6]{x}\) also increases from 0 to \(\infty\). Thus, the range of \(f(x)\) is \(f(x) \geq 0\), or \([0, \infty)\). The graph starts at the origin \((0,0)\) and passes through points like \((1,1)\) and \((64,2)\), displaying a curve that increases at a decreasing rate.
For the function \(h(x) = x^{1/7} = \sqrt[7]{x}\), we observe that it involves an odd root. Odd roots are well-defined for all real numbers because a negative number raised to an odd power remains negative. Therefore, the domain of \(h(x)\) is all real numbers, denoted as \((-\infty, \infty)\). Similarly, the range of \(h(x)\) extends over all real numbers as \(x\) varies from \(-\infty\) to \(\infty\). The graph passes through the origin \((0,0)\), \((1,1)\), and \((-1,-1)\). It also passes through \((128,2)\) and \((-128,-2)\), creating a shape that is symmetric with respect to the origin and increases throughout its entire domain.
The absolute value of a number \(a\), denoted as \(|a|\), represents its distance from 0 on the number line. It is defined as \(|a| = a\) if \(a \geq 0\) and \(|a| = -a\) if \(a < 0\).
In everyday life, functions like square roots and cube roots are used in geometry to relate areas and volumes to side lengths. For example, if you know the volume of a cube is \(V\), its side length is found by calculating \(V^{1/3}\). Similarly, absolute values are essential in situations where direction doesn't matter, such as measuring the distance between two points on a map or calculating the margin of error in manufacturing or data collection, ensuring that variations are always treated as positive magnitudes.