Which of the following functions are odd? Select all that apply
f(x)=x/x^8+8x^4-7
f(x)=x^3+4x
f(x)=3x^5+6x^4+4
f(x)=8x^2-8
A function is considered odd if it satisfies the property f(-x) = -f(x) for all x in the domain of the function.
Let's check each function:
f(x) = x/x^8 + 8x^4 - 7
Since the function does not have a clear negative sign term or an even exponent on x, it is not odd.
f(x) = x^3 + 4x
We have:
f(-x) = (-x)^3 + 4(-x) = -x^3 - 4x
-f(x) = -(x^3 + 4x) = -x^3 - 4x
Since f(-x) = -f(x), this function is odd.
f(x) = 3x^5 + 6x^4 + 4
Since the function does not have a clear negative sign term or an even exponent on x, it is not odd.
f(x) = 8x^2 - 8
We have:
f(-x) = 8(-x)^2 - 8 = 8x^2 - 8
-f(x) = -(8x^2 - 8) = -8x^2 + 8
Since f(-x) = -f(x), this function is odd.
The odd functions are:
f(x) = x^3 + 4x
f(x) = 8x^2 - 8
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 go through each function and check:
1. f(x) = x/x^8 + 8x^4 - 7
To test for oddness, we substitute -x into the function:
f(-x) = (-x)/(-x)^8 + 8(-x)^4 - 7
= -x/x^8 + 8x^4 - 7
= -(x/x^8 + 8x^4 - 7)
So, it satisfies the property f(-x) = -f(x).
2. f(x) = x^3 + 4x
Testing for oddness, let's substitute -x into the function:
f(-x) = (-x)^3 + 4(-x)
= -x^3 - 4x
This is not equal to -(x^3 + 4x), so it does not satisfy the property f(-x) = -f(x).
3. f(x) = 3x^5 + 6x^4 + 4
Substituting -x into the function:
f(-x) = 3(-x)^5 + 6(-x)^4 + 4
= -3x^5 + 6x^4 + 4
This is not equal to -(3x^5 + 6x^4 + 4), so it does not satisfy the property f(-x) = -f(x).
4. f(x) = 8x^2 - 8
Substituting -x into the function:
f(-x) = 8(-x)^2 - 8
= 8x^2 - 8
This is equal to f(x), but it's not equal to -(8x^2 - 8), so it does not satisfy the property f(-x) = -f(x).
Based on our analysis, only function f(x) = x/x^8 + 8x^4 - 7 is odd.
In order to determine if a function is odd, we need to check if f(-x) = -f(x) for all x in the domain of the function.
Let's go through each function and perform this check:
1) f(x) = x/x^8 + 8x^4 - 7
To check if it is odd, we need to determine f(-x). Let's substitute -x into the function:
f(-x) = (-x) / (-x)^8 + 8(-x)^4 - 7
= -x / x^8 + 8x^4 - 7
Now let's check if -f(x) is equal to f(-x):
-f(x) = -x / x^8 + 8x^4 - 7
Since -f(x) = f(-x), we can conclude that the function f(x) = x/x^8 + 8x^4 - 7 is odd.
2) f(x) = x^3 + 4x
For this function, let's check f(-x):
f(-x) = (-x)^3 + 4(-x)
= -x^3 - 4x
Now let's check if -f(x) is equal to f(-x):
-f(x) = -x^3 - 4x
Since -f(x) = f(-x), we can conclude that the function f(x) = x^3 + 4x is odd.
3) f(x) = 3x^5 + 6x^4 + 4
Let's check f(-x):
f(-x) = 3(-x)^5 + 6(-x)^4 + 4
= -3x^5 + 6x^4 + 4
Now let's check if -f(x) is equal to f(-x):
-f(x) = -3x^5 - 6x^4 - 4
Since -f(x) is not equal to f(-x), we can conclude that the function f(x) = 3x^5 + 6x^4 + 4 is not odd.
4) f(x) = 8x^2 - 8
For this function, let's check f(-x):
f(-x) = 8(-x)^2 - 8
= 8x^2 - 8
Now let's check if -f(x) is equal to f(-x):
-f(x) = -8x^2 + 8
Since -f(x) is not equal to f(-x), we can conclude that the function f(x) = 8x^2 - 8 is not odd.
To summarize, the functions that are odd are:
- f(x) = x/x^8 + 8x^4 - 7
- f(x) = x^3 + 4x
The functions f(x) = 3x^5 + 6x^4 + 4 and f(x) = 8x^2 - 8 are not odd.