Which of the following functions are odd? Select all that apply.
(1 point)
Responses
f(x) = 7x^5-4x
f(x)=3x^2-9x
f(x)=6x^7+4x^3-2
f(x)=-5x^9+8x^5+4x^3
f(x)=2x^3+5
A function is odd if f(-x) = -f(x) for all values of x.
The functions f(x) = 7x^5-4x and f(x)=2x^3+5 are odd functions.
f(-x) = 7(-x)^5-4(-x) = -7x^5 + 4x = -f(x)
f(-x) = 2(-x)^3 + 5 = -2x^3 + 5 = -f(x)
Therefore, the functions f(x) = 7x^5-4x and f(x)=2x^3+5 are odd.
To determine if a function is odd, we need to check if the function satisfies the property f(-x) = -f(x) for all x in the domain.
Let's check each function one by one:
1. For f(x) = 7x^5 - 4x,
-f(x) = -7x^5 + 4x.
Since -f(x) is not equal to f(-x), this function is not odd.
2. For f(x) = 3x^2 - 9x,
-f(x) = -3x^2 + 9x.
Since -f(x) is not equal to f(-x), this function is not odd.
3. For f(x) = 6x^7 + 4x^3 - 2,
-f(x) = -6x^7 - 4x^3 + 2.
Since -f(x) is equal to f(-x), this function is odd.
4. For f(x) = -5x^9 + 8x^5 + 4x^3,
-f(x) = 5x^9 - 8x^5 - 4x^3.
Since -f(x) is equal to f(-x), this function is odd.
5. For f(x) = 2x^3 + 5,
-f(x) = -2x^3 - 5.
Since -f(x) is equal to f(-x), this function is odd.
Therefore, the odd functions are:
- f(x) = 6x^7 + 4x^3 - 2
- f(x) = -5x^9 + 8x^5 + 4x^3
- f(x) = 2x^3 + 5
To determine whether a function is odd, you need to check if the function satisfies the property of odd functions. An odd function is symmetric about the origin, meaning that if you reflect the function about the origin, it remains unchanged.
To check if a function is odd, you can use the property that for an odd function, f(-x) = -f(x).
Let's go through each function and apply this property to determine if they are odd:
1. f(x) = 7x^5 - 4x
To check if this function is odd, we need to evaluate f(-x) and see if it equals -f(x):
f(-x) = 7(-x)^5 - 4(-x) = -7x^5 + 4x
-f(x) = -(7x^5 - 4x) = -7x^5 + 4x
Since f(-x) = -f(x), this function is odd.
2. f(x) = 3x^2 - 9x
Checking if this function is odd:
f(-x) = 3(-x)^2 - 9(-x) = 3x^2 + 9x
-f(x) = -(3x^2 - 9x) = -3x^2 + 9x
Since f(-x) ≠ -f(x), this function is not odd.
3. f(x) = 6x^7 + 4x^3 - 2
Checking if this function is odd:
f(-x) = 6(-x)^7 + 4(-x)^3 - 2 = -6x^7 - 4x^3 - 2
-f(x) = -(6x^7 + 4x^3 - 2) = -6x^7 - 4x^3 + 2
Since f(-x) ≠ -f(x), this function is not odd.
4. f(x) = -5x^9 + 8x^5 + 4x^3
Checking if this function is odd:
f(-x) = -5(-x)^9 + 8(-x)^5 + 4(-x)^3 = -5x^9 + 8x^5 - 4x^3
-f(x) = -(-5x^9 + 8x^5 + 4x^3) = 5x^9 - 8x^5 - 4x^3
Since f(-x) ≠ -f(x), this function is not odd.
5. f(x) = 2x^3 + 5
Checking if this function is odd:
f(-x) = 2(-x)^3 + 5 = -2x^3 + 5
-f(x) = -(2x^3 + 5) = -2x^3 - 5
Since f(-x) ≠ -f(x), this function is not odd.
Based on the analysis, the only function that is odd is f(x) = 7x^5 - 4x.