Question

question 10 (multiple choice worth 5 points) (03.02r mc) which statement is true about the function f(x) = 6x⁷? the function is even because f(−x) = f(x). the function is odd because f(−x) = −f(x). the function is odd because f(−x) = f(x). the function is even because f(−x) = −f(x).

Explanation:

Step1: Recall definitions of even and odd functions

An even function satisfies \( f(-x) = f(x) \) for all \( x \) in the domain. An odd function satisfies \( f(-x) = -f(x) \) for all \( x \) in the domain.

Step2: Compute \( f(-x) \) for \( f(x) = 6x^7 \)

Substitute \( -x \) into the function: \( f(-x) = 6(-x)^7 \). Since \( (-x)^7 = -x^7 \) (because 7 is odd), we have \( f(-x) = 6(-x^7) = -6x^7 \).

Step3: Compare \( f(-x) \) with \( -f(x) \)

Compute \( -f(x) \): \( -f(x) = - (6x^7) = -6x^7 \). So \( f(-x) = -f(x) \), which means the function is odd.

Answer:

The function is odd because \( f(-x) = -f(x) \). (The option corresponding to this statement, e.g., if the third option is "The function is odd because \( f(-x) = -f(x) \)", then the answer is that option with its label)