Question

The function ƒ is given by fx=x³+x . Which of the following statements is true and supports the claim that f is an odd function and not an even function?

A) f (0) =-f (0)
B) -f(3) = f(3)
C) f(-3) =f (3)
D) f(-3) =-f (3)

Answer 1

The function given is f(x)=x³+x. To determine if it is an odd function, we evaluate f(-3) and -f(3). Since f(-3)=-33 and -f(3)=-30, we can conclude that f is an odd function. Therefore, the correct answer is D).

Step-by-step explanation:

The function ƒ is given by f(x)=x³+x. To determine whether f is an odd function or an even function, we need to check certain properties:

1. An odd function satisfies the property f(-x)=-f(x).
2. An even function satisfies the property f(-x)=f(x).

Let's evaluate the function at x=0:

• f(0)=0³+0=0. Now let's check if f(-0)=-f(0):
• f(-0)=(-0)³+(-0)=0. Since f(-0)=f(0) and not -f(0), this indicates that the function f is not an odd function. Therefore, the correct answer is not A). Now let's examine the other options:
• -f(3)=-(3³+3)=-30, but f(3)=3³+3=33. Therefore, B) is incorrect.
• f(-3)=(-3)³+(-3)=-33, but f(3)=3³+3=33. Therefore, C) is incorrect.
• f(-3)=(-3)³+(-3)=-33, but -f(3)=-(3³+3)=-30. Therefore, D) is true and supports the claim that f is an odd function and not an even function. Therefore, the correct answer is D).