What is the range of \( f(x)=\sqrt[3]{x-4} \) ? (1 point) \( \{y|y \in \mathbb{R}|\} \) \( \{x \mid-\infty

Ask by Burgess Wood. in the United States

Jan 23,2025

The function \( f(x) = \sqrt[3]{x - 4} \) is a cube root function, which means it can take any real number as its input. Since the cube root is defined for all real numbers, as \( x \) approaches negative infinity, \( f(x) \) will approach negative infinity. Conversely, as \( x \) approaches positive infinity, \( f(x) \) will also approach positive infinity. Therefore, the range of \( f(x) \) is indeed all real numbers: \( \{y | y \in \mathbb{R}\} \). If you were to graph this function, you’d see a smooth curve crossing through the point \( (4, 0) \) — that's where the input makes the expression inside the cube root zero. Beyond that point, it gracefully extends downwards to the left and upwards to the right, portraying its infinite range beautifully!