Which functions are odd? Select all that apply.
(2 points)
Responses
f left-parenthesis x right-parenthesis equals negative one-half x superscript 4 baseline plus 5
Image with alt text: f left-parenthesis x right-parenthesis equals negative one-half x superscript 4 baseline plus 5
f left-parenthesis x right-parenthesis equals negative 8 x cubed plus 5 x
Image with alt text: f left-parenthesis x right-parenthesis equals negative 8 x cubed plus 5 x
The odd functions are functions that satisfy the property f(x) = -f(-x).
Only the second function, f(x) = -8x^3 + 5x, satisfies this property and is therefore odd.
To determine whether a function is odd, we need to check if it satisfies the property f(-x) = -f(x) for all x in its domain. Let's check each function:
1. f(x) = -1/2x^4 + 5
To check if this function is odd, we substitute -x for x and simplify:
f(-x) = -1/2(-x)^4 + 5
= -1/2x^4 + 5
Since f(-x) = f(x) for this function, it is an even function.
2. f(x) = -8x^3 + 5x
To check if this function is odd, we substitute -x for x and simplify:
f(-x) = -8(-x)^3 + 5(-x)
= 8x^3 - 5x
Now, let's check if -f(x) is equal to f(-x):
-f(x) = -(-8x^3 + 5x)
= 8x^3 - 5x
Since -f(x) = f(-x) for this function, it is an odd function.
So, the function f(x) = -8x^3 + 5x is odd.