Which of the following functions are odd? Select all that apply. (1 point) Responses Image with alt text: f left parenthesis x right parenthesis equals 7 x superscript 5 baseline minus 4 x Image with alt text: f left parenthesis x right parenthesis equals 3 x squared minus 9 x Image with alt text: f left parenthesis x right parenthesis equals 6 x superscript 7 baseline plus 4 x cubed minus 2 Image with alt text: f left parenthesis x right parenthesis equals negative 5 x superscript 9 baseline plus 8 x superscript 5 baseline plus 4 x cubed
The odd functions are the functions that satisfy f(-x) = -f(x).
Among the given functions, the odd functions are:
1. f(x) = 7x^5 - 4x (because f(-x) = 7(-x)^5 - 4(-x) = -7x^5 + 4x)
2. f(x) = 6x^7 + 4x^3 - 2 (because f(-x) = 6(-x)^7 + 4(-x)^3 - 2 = -6x^7 - 4x^3 - 2)
Therefore, the odd functions are f(x) = 7x^5 - 4x and f(x) = 6x^7 + 4x^3 - 2.
To determine if a function is odd, we need to check if the function satisfies the property f(x) = -f(-x) for all values of x in the domain.
Let's check each function one by one:
1. f(x) = 7x^5 - 4x
To check if it is odd, we substitute -x into the function:
f(-x) = 7(-x)^5 - 4(-x)
= -7x^5 + 4x
Now comparing f(x) and -f(-x):
- f(-x) = -(-7x^5 + 4x) = 7x^5 - 4x
Since f(x) = -f(-x), this function is odd.
2. f(x) = 3x^2 - 9x
Substituting -x into the function:
f(-x) = 3(-x)^2 - 9(-x)
= 3x^2 + 9x
Comparing f(x) and -f(-x):
- 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
Substituting -x into the function:
f(-x) = 6(-x)^7 + 4(-x)^3 - 2
= -6x^7 + 4x^3 - 2
Comparing f(x) and -f(-x):
- f(-x) = -(-6x^7 + 4x^3 - 2) = 6x^7 - 4x^3 + 2
Since f(x) = -f(-x), this function is odd.
4. f(x) = -5x^9 + 8x^5 + 4x^3
Substituting -x into the function:
f(-x) = -5(-x)^9 + 8(-x)^5 + 4(-x)^3
= -5x^9 + 8x^5 - 4x^3
Comparing f(x) and -f(-x):
- f(-x) = -(-5x^9 + 8x^5 - 4x^3) = 5x^9 - 8x^5 + 4x^3
Since f(x) = -f(-x), this function is odd.
In conclusion, the odd functions are:
- f(x) = 7x^5 - 4x
- f(x) = 6x^7 + 4x^3 - 2
- f(x) = -5x^9 + 8x^5 + 4x^3
To determine whether a function is odd, we can use the property that an odd function satisfies the condition: f(-x) = -f(x) for all x in the domain of the function.
Let's check each function to see if it satisfies this property:
1. f(x) = 7x^5 - 4x
To check if it's odd, we substitute -x into the function:
f(-x) = 7(-x)^5 - 4(-x)
= -7x^5 + 4x
Comparing this with the original function, f(x), we can see that -f(x) = -7x^5 + 4x.
Since f(-x) = -f(x), this function is odd.
2. f(x) = 3x^2 - 9x
Similarly, we substitute -x into the function:
f(-x) = 3(-x)^2 - 9(-x)
= 3x^2 + 9x
Comparing this with the original function, f(x), we can see that -f(x) = -3x^2 + 9x.
Since f(-x) = -f(x), this function is odd.
3. f(x) = 6x^7 + 4x^3 - 2
Substituting -x into the function:
f(-x) = 6(-x)^7 + 4(-x)^3 - 2
= -6x^7 - 4x^3 - 2
Comparing this with the original function, f(x), we can see that -f(x) = -6x^7 - 4x^3 + 2.
Since f(-x) = -f(x) in this case, this function is odd.
4. f(x) = -5x^9 + 8x^5 + 4x^3
Substituting -x into the function:
f(-x) = -5(-x)^9 + 8(-x)^5 + 4(-x)^3
= -5x^9 + 8x^5 - 4x^3
Comparing this with the original function, f(x), we can see that -f(x) = 5x^9 - 8x^5 - 4x^3.
Since f(-x) is not equal to -f(x), this function is not odd.
So, the odd functions from the given options are:
- f(x) = 7x^5 - 4x
- f(x) = 3x^2 - 9x
- f(x) = 6x^7 + 4x^3 - 2