Which of the following functions are odd? Select all that apply.
a. f(×)=7×^5-4×
b. f(×)=3×^2-9×
c. f(×)=6×^7+4×^3-2
d. f(×)=-5×^9+8×^5+4×^3
e. f(×)=2×^3+5
A function is odd if and only if f(-x) = -f(x) for every x in the domain of f.
a. f(x)=7x^5-4x
To check if it is odd, we substitute -x for x:
f(-x) = 7(-x)^5-4(-x) = -7x^5+4x
This is not equal to -f(x) = -7x^5+4x.
So, this function is not odd.
b. f(x)=3x^2-9x
To check if it is odd, we substitute -x for x:
f(-x) = 3(-x)^2-9(-x) = 3x^2+9x
This is not equal to -f(x) = -(3x^2-9x) = -3x^2+9x.
So, this function is not odd.
c. f(x)=6x^7+4x^3-2
To check if it is odd, we substitute -x for x:
f(-x) = 6(-x)^7+4(-x)^3-2 = -6x^7-4x^3-2
This is equal to -f(x) = -6x^7-4x^3-2.
So, this function is odd.
d. f(x)=-5x^9+8x^5+4x^3
To check if it is odd, we substitute -x for x:
f(-x) = -5(-x)^9+8(-x)^5+4(-x)^3 = -5x^9-8x^5-4x^3
This is equal to -f(x) = -(-5x^9+8x^5+4x^3).
So, this function is odd.
e. f(x)=2x^3+5
To check if it is odd, we substitute -x for x:
f(-x) = 2(-x)^3+5 = -2x^3+5
This is not equal to -f(x) = -(2x^3+5) = -2x^3-5.
So, this function is not odd.
Therefore, the odd functions are c. f(x)=6x^7+4x^3-2 and d. f(x)=-5x^9+8x^5+4x^3.
To determine whether a function is odd, we need to check if it satisfies the condition -f(x) = f(-x) for all values of x in the domain of the function.
Let's go through each option and check:
a. f(x) = 7x^5 - 4x
To check if this function is odd, we need to substitute -x instead of x and see if the equation holds:
-f(x) = -[7(-x)^5 - 4(-x)] = -[-7x^5 + 4x] = 7x^5 - 4x
This is equal to f(-x). Therefore, option (a) is odd.
b. f(x) = 3x^2 - 9x
-f(x) = -[3(-x)^2 - 9(-x)] = -[3x^2 + 9x] = -3x^2 - 9x
This is not equal to f(-x). Therefore, option (b) is not odd.
c. f(x) = 6x^7 + 4x^3 - 2
-f(x) = -[6(-x)^7 + 4(-x)^3 - 2] = -[6x^7 + 4x^3 - 2] = -6x^7 - 4x^3 + 2
This is not equal to f(-x). Therefore, option (c) is not odd.
d. f(x) = -5x^9 + 8x^5 + 4x^3
-f(x) = -[-5(-x)^9 + 8(-x)^5 + 4(-x)^3] = -[-5x^9 + 8x^5 + 4x^3]
This is equal to f(-x). Therefore, option (d) is odd.
e. f(x) = 2x^3 + 5
-f(x) = -[2(-x)^3 + 5] = -[2x^3 + 5]
This is not equal to f(-x). Therefore, option (e) is not odd.
In summary, the odd functions are:
a. f(x) = 7x^5 - 4x
d. f(x) = -5x^9 + 8x^5 + 4x^3
To determine if a function is odd, we need to check if f(-x) is equal to -f(x) for all values of x.
a. f(x) = 7x^5 - 4x
Checking f(-x): f(-x) = 7(-x)^5 - 4(-x) = -7x^5 + 4x
Checking -f(x): -f(x) = -(7x^5 - 4x) = -7x^5 + 4x
Since f(-x) = -f(x), function a is odd.
b. f(x) = 3x^2 - 9x
Checking f(-x): f(-x) = 3(-x)^2 - 9(-x) = 3x^2 + 9x
Checking -f(x): -f(x) = -(3x^2 - 9x) = -3x^2 + 9x
Since f(-x) is not equal to -f(x), function b is not odd.
c. f(x) = 6x^7 + 4x^3 - 2
Checking f(-x): f(-x) = 6(-x)^7 + 4(-x)^3 - 2 = -6x^7 - 4x^3 - 2
Checking -f(x): -f(x) = -(6x^7 + 4x^3 - 2) = -6x^7 - 4x^3 + 2
Since f(-x) is not equal to -f(x), function c is not odd.
d. f(x) = -5x^9 + 8x^5 + 4x^3
Checking f(-x): f(-x) = -5(-x)^9 + 8(-x)^5 + 4(-x)^3 = -5x^9 + 8x^5 - 4x^3
Checking -f(x): -f(x) = -(-5x^9 + 8x^5 + 4x^3) = 5x^9 - 8x^5 - 4x^3
Since f(-x) is not equal to -f(x), function d is not odd.
e. f(x) = 2x^3 + 5
Checking f(-x): f(-x) = 2(-x)^3 + 5 = -2x^3 + 5
Checking -f(x): -f(x) = -(2x^3 + 5) = -2x^3 - 5
Since f(-x) = -f(x), function e is odd.
Therefore, the odd functions are a and e.