Identify the domain and range of the following radical function. \( y=\sqrt[3]{x}-5 \) Domain: All Real Numbers Range: All Real Numbers Domain: \( x \geq 5 \) Range: \( y \geq 1 \) Domain: \( x \geq 0 \) Range: \( y \geq 1 \) Domain: \( x \geq 1 \) Range: \( y \geq 5 \)

The function \( y=\sqrt[3]{x}-5 \) is a radical function with a cube root, which means it can take on all real numbers for both its input (x-values) and output (y-values). Why is that? Well, cube roots can handle positive, negative, and zero values seamlessly! Therefore, the domain is indeed all real numbers, and naturally, since we can produce any real number output by adjusting \( x \), the range is also all real numbers. If you're exploring this in more depth, think about what happens if you fiddle with the 5 at the end. Shifting variations up or down in graphing play a huge role! For instance, if you added a constant instead of subtracting 5, it would raise the entire graph. Such shifts are critical in understanding transformations of functions. Having fun with those transformations can give you unique insights into behavior as you explore other functions too!