Question

given the function $f(x) = -x^3 + x$, which of the following expressions are equivalent to $f(-x)$? select all that apply. $\square\\ x^3 - x$ $\square\\ -x^3 + x$ $\square\\ -x^3 - x$ $\square\\ -f(x)$ $\square\\ f(x)$ $\square\\ -(-x^3 + x)$

Explanation:

Step1: Find \( f(-x) \)

Substitute \( -x \) into \( f(x) = -x^3 + x \). So \( f(-x) = -(-x)^3 + (-x) \).

Step2: Simplify \( (-x)^3 \)

We know that \( (-x)^3 = -x^3 \), so \( -(-x)^3 = -(-x^3) = x^3 \). Then \( f(-x) = x^3 - x \).

Step3: Analyze \( -f(x) \)

First, find \( -f(x) \). Since \( f(x) = -x^3 + x \), then \( -f(x) = -(-x^3 + x) = x^3 - x \), which is the same as \( f(-x) \). Also, \( -(-x^3 + x) \) is equal to \( x^3 - x \) as we saw from simplifying \( -f(x) \). The option \( x^3 - x \) is the simplified form of \( f(-x) \), \( -f(x) \) simplifies to \( x^3 - x \), and \( -(-x^3 + x) \) also simplifies to \( x^3 - x \). The other options: \( -x^3 + x \) is \( f(x) \) not \( f(-x) \), \( -x^3 - x \) does not match our calculation of \( f(-x) \), and \( f(x) \) is not equal to \( f(-x) \) here.

Answer:

\( x^3 - x \), \( -f(x) \), \( -(-x^3 + x) \) (corresponding to the options: the first option \( x^3 - x \), the fourth option \( -f(x) \), the sixth option \( -(-x^3 + x) \))