Question

true or false? the function $k(x) = \frac{1}{3}x^3$ and the inverse $k^{-1}(x)$ are symmetric over the line $y = x$
option #1 true
option #2 false
(1 point)

Explanation:

Step1: Recall Inverse Function Symmetry

A function and its inverse are symmetric about the line \( y = x \). This is a fundamental property of inverse functions.

Step2: Analyze the Given Function

The function \( k(x)=\frac{1}{3}x^3 \) is a cubic function. To find its inverse, we swap \( x \) and \( y \) (let \( y = k(x) \)): \( x=\frac{1}{3}y^3 \), then solve for \( y \): \( y^3 = 3x \), so \( y = \sqrt[3]{3x} \), which is \( k^{-1}(x) \).

Step3: Check Symmetry with Graph

From the graph, we can visually inspect the symmetry of \( k(x) \) (red curve) and its inverse (dashed curves) about \( y = x \) (the dotted line \( y = x \) is also shown, likely the line of symmetry). The curves mirror each other across \( y = x \), consistent with the inverse function symmetry property.

Answer:

Option #1 True