Answer:
D) [tex]f(x)=8x^5+7x^3-5x[/tex]
Step-by-step explanation:
A function is odd if its graph is symmetric to the origin, which we can check this if [tex]f(-x)=-f(x)[/tex] is true:
Option A
[tex]f(x)=7x^4+13x^3-11\\\\f(-x)=7(-x)^4+13(-x)^3-11\\\\f(-x)=7x^4-13x^3-11[/tex]
Since [tex]f(-x)\neq-f(x)[/tex], then the function does not have odd symmetry
Option B
[tex]f(x)=5x^7+4x^6+3x^5+x\\\\f(-x)=5(-x)^7+4(-x)^6+3(-x)^5+x\\\\f(-x)=-5x^7+4x^6-3x^5+x[/tex]
Since [tex]f(-x)\neq-f(x)[/tex], then the function does not have odd symmetry
Option C
[tex]f(x)=13x^5-\frac{1}{2}x^3+7x+3\\\\f(-x)=13(-x)^5-\frac{1}{2}(-x)^4+7(-x)+3\\ \\f(-x)=-13x^5-\frac{1}{2}x^4-7x+3[/tex]
Since [tex]f(-x)\neq-f(x)[/tex], then the function does not have odd symmetry
Option D
[tex]f(x)=8x^5+7x^3-5x\\\\f(-x)=8(-x)^5+7(-x)^3-5(-x)\\\\f(-x)=-8x^5-7x^3+5x[/tex]
Since [tex]f(-x)=-f(x)[/tex], then the function DOES have odd symmetry. You can also see that the function is odd because every term has an odd exponent.
