which of the following functions are odd select all that apply
A f(x)=7x^5-4x
B f(x)=3x^2-9x
C f(x)=6x^7+4x^3-2
D f(x)=-5x^9+8x^5+4x^3
E f(x)2x^3+5
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:
A) f(x) = 7x^5 - 4x
-f(x) = -7x^5 + 4x
f(-x) = 7(-x)^5 - 4(-x) = -7x^5 + 4x
f(-x) = -f(x) = -7x^5 + 4x for all x
Hence, function A is odd.
B) f(x) = 3x^2 - 9x
-f(x) = -3x^2 + 9x
f(-x) = 3(-x)^2 - 9(-x) = 3x^2 + 9x
f(-x) is not equal to -f(x) for all x
Hence, function B is not odd.
C) f(x) = 6x^7 + 4x^3 - 2
-f(x) = -6x^7 - 4x^3 + 2
f(-x) = 6(-x)^7 + 4(-x)^3 - 2 = -6x^7 - 4x^3 - 2
f(-x) = -f(x) = -6x^7 - 4x^3 - 2 for all x
Hence, function C is odd.
D) f(x) = -5x^9 + 8x^5 + 4x^3
-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
f(-x) = -f(x) = -5x^9 + 8x^5 - 4x^3 for all x
Hence, function D is odd.
E) f(x) = 2x^3 + 5
-f(x) = -2x^3 - 5
f(-x) = 2(-x)^3 + 5 = -2x^3 + 5
f(-x) is not equal to -f(x) for all x
Hence, function E is not odd.
The functions that are odd are A and C.
A function is odd if f(-x) = -f(x) for all x in the domain.
Let's check each function one by one:
A) f(x) = 7x^5 - 4x
f(-x) = 7(-x)^5 - 4(-x) = -7x^5 + 4x
-f(x) = -7x^5 + 4x
Since f(-x) = -f(x), this function is odd.
B) f(x) = 3x^2 - 9x
f(-x) = 3(-x)^2 - 9(-x) = 3x^2 + 9x
-f(x) = -3x^2 + 9x
Since f(-x) is not equal to -f(x), this function 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
-f(x) = -6x^7 - 4x^3 + 2
Since f(-x) = -f(x), this function is 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
-f(x) = 5x^9 - 8x^5 - 4x^3
Since f(-x) is not equal to -f(x), this function is not odd.
E) f(x) = 2x^3 + 5
f(-x) = 2(-x)^3 + 5 = -2x^3 + 5
-f(x) = -2x^3 - 5
Since f(-x) is not equal to -f(x), this function is not odd.
So, the odd functions are A) f(x) = 7x^5 - 4x and C) f(x) = 6x^7 + 4x^3 - 2.
To determine whether a function is odd, we need to check if it satisfies the property of odd functions, which is f(-x) = -f(x) for all x in the domain of the function.
Let's check each function one by one:
A) f(x) = 7x^5 - 4x
To check if it's odd, we need to compute f(-x) and compare it to -f(x).
f(-x) = 7(-x)^5 - 4(-x) = -7x^5 + 4x
-f(x) = -7x^5 + 4x
Since f(-x) = -f(x), function A is odd.
B) f(x) = 3x^2 - 9x
f(-x) = 3(-x)^2 - 9(-x) = 3x^2 + 9x
-f(x) = -3x^2 + 9x
Since f(-x) does not equal -f(x), function 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
-f(x) = -6x^7 - 4x^3 - 2
Since f(-x) = -f(x), function C is 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
-f(x) = 5x^9 - 8x^5 - 4x^3
Since f(-x) does not equal -f(x), function D is not odd.
E) f(x) = 2x^3 + 5
f(-x) = 2(-x)^3 + 5 = -2x^3 + 5
-f(x) = -2x^3 - 5
Since f(-x) does not equal -f(x), function E is not odd.
Based on the above analysis, the odd functions are:
A) f(x) = 7x^5 - 4x
C) f(x) = 6x^7 + 4x^3 - 2