Question

and cube root functions and equations
which is the graph of the cube root function $f(x) = \sqrt3{x}$?

Explanation:

Step1: Analyze cube root properties

The cube root function $f(x)=\sqrt[3]{x}$ has the following key properties:

• Domain and range are all real numbers ($(-\infty, \infty)$).
• When $x>0$, $f(x)>0$; when $x<0$, $f(x)<0$.
• The function is strictly increasing across its entire domain.
• It passes through the origin $(0,0)$.

Step2: Evaluate each graph

1. Top graph: Decreasing (as $x$ increases, $y$ decreases) → does not match the increasing nature of $f(x)=\sqrt[3]{x}$.
2. Second graph: Increasing, $x>0$ gives $y>0$, $x<0$ gives $y<0$, passes through $(0,0)$ → matches all properties.
3. Third graph: Has a steeper, non-cube-root curve shape (resembles a linear or square root curve, not the characteristic S-shape of a cube root) → does not match.
4. Bottom graph: Decreasing (as $x$ increases, $y$ decreases) → does not match the increasing nature of $f(x)=\sqrt[3]{x}$.

Answer:

The second graph (the one with an upward-sloping S-shape, passing through the origin, where positive x-values correspond to positive y-values and negative x-values correspond to negative y-values)