Question

the function $g(x) = x^3 - 4$ and its inverse are even functions.
option #1: true
option #2: false
(1 point)

Explanation:

Step1: Recall even function definition

A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in its domain.

Step2: Check \( g(x) = x^3 - 4 \)

Compute \( g(-x) = (-x)^3 - 4 = -x^3 - 4 \). Compare with \( g(x) = x^3 - 4 \). Since \( -x^3 - 4
eq x^3 - 4 \) (e.g., \( x = 1 \): \( g(-1) = -1 - 4 = -5 \), \( g(1) = 1 - 4 = -3 \); \( -5
eq -3 \)), \( g(x) \) is not even.

Step3: Inverse function property

The inverse of a non - even function (cubic function \( g(x) = x^3 - 4 \) is odd - like shifted) will also not be even. Also, from the graph, \( g(x) \) and \( g^{-1}(x) \) are symmetric about \( y = x \), not about the y - axis (even function symmetry).

Answer:

Option #2: False