Respuesta :
An odd function is such that: f(-x) = -f(x)
[tex]\mathbf{(a)\ f(x) = 7x^5 - 4x}[/tex] and [tex]\mathbf{(d)\ f(x) = -5x^9 + 8x^5 + 4x}[/tex] are odd functions
[tex]\mathbf{(a)\ f(x) = 7x^5 - 4x}[/tex]
Calculate f(-x)
[tex]\mathbf{f(-x) = 7(-x)^5 - 4(-x)}[/tex]
[tex]\mathbf{f(-x) = -7x^5 + 4x}[/tex]
Calculate -f(x)
[tex]\mathbf{-f(x) = -(7x^5 - 4x)}[/tex]
[tex]\mathbf{-f(x) = -7x^5 + 4x}[/tex]
[tex]\mathbf{f(x) = 7x^5 - 4x}[/tex] is odd, because f(-x) = -f(x)
[tex]\mathbf{(b)\ f(x) = 3x^2 - 9x}[/tex]
Calculate f(-x)
[tex]\mathbf{f(-x) = 3(-x)^2 - 9(-x)}[/tex]
[tex]\mathbf{f(-x) = 3x^2 + 9x}[/tex]
Calculate -f(x)
[tex]\mathbf{-f(x) = -(3x^2 - 9x)}[/tex]
[tex]\mathbf{-f(x) = -3x^2 + 9x}[/tex]
[tex]\mathbf{(b)\ f(x) = 3x^2 - 9x}[/tex] is not odd, because f(-x) != -f(x)
[tex]\mathbf{(c)\ f(x) = 6x^7 + 4x^3 - 2}[/tex]
Calculate f(-x)
[tex]\mathbf{f(-x) = 6(-x)^7 + 4(-x)^3 - 2}[/tex]
[tex]\mathbf{f(-x) = -6x^7 - 4x^3 - 2}[/tex]
Calculate -f(x)
[tex]\mathbf{-f(-x) = -(6x^7 + 4x^3 - 2)}[/tex]
[tex]\mathbf{-f(-x) = -6x^7 - 4x^3 + 2}[/tex]
[tex]\mathbf{(c)\ f(x) = 6x^7 + 4x^3 - 2}[/tex] is not odd, because f(-x) != -f(x)
[tex]\mathbf{(d)\ f(x) = -5x^9 + 8x^5 + 4x}[/tex]
Calculate f(-x)
[tex]\mathbf{f(-x) = -5(-x)^9 + 8(-x)^5 + 4(-x)}[/tex]
[tex]\mathbf{f(-x) = 5x^9 - 8x^5 - 4x}[/tex]
Calculate -f(x)
[tex]\mathbf{-f(x) = -(-5x^9 + 8x^5 + 4x)}[/tex]
[tex]\mathbf{-f(x) = 5x^9 - 8x^5 - 4x}[/tex]
[tex]\mathbf{(d)\ f(x) = -5x^9 + 8x^5 + 4x}[/tex] is odd, because f(-x) = -f(x)
[tex]\mathbf{(e)\ f(x) = 2x^3 +5}[/tex]
Calculate f(-x)
[tex]\mathbf{f(-x) = 2(-x)^3 +5}[/tex]
[tex]\mathbf{f(-x) = -2x^3 +5}[/tex]
Calculate -f(x)
[tex]\mathbf{-f(x) = -(2x^3 +5)}[/tex]
[tex]\mathbf{-f(x) = -2x^3 -5}[/tex]
[tex]\mathbf{(e)\ f(x) = 2x^3 +5}[/tex] is not odd, because f(-x) != -f(x)
Hence, the odd functions are:
[tex]\mathbf{(a)\ f(x) = 7x^5 - 4x}[/tex] and [tex]\mathbf{(d)\ f(x) = -5x^9 + 8x^5 + 4x}[/tex]
Read more about odd functions at:
https://brainly.com/question/15775372
