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Which function is an odd function?
A. f(x) = cos(-2pi x)
B. f(x) =csc(-3pi/2 x)
C. f(x) = sec (5pi/3 x)
D. f(x) = cos(6pi/5 x)

Sagot :

Answer:

B. f(x) =csc(-3pi/2 x)

Step-by-step explanation:

Got it right on Edmentum.

The function [tex]f(x)=csc(-\frac{3\pi}{2x} )[/tex] is an odd function.

The correct answer is an option (B)

What is function?

  • "It defines a relation between input and output values."
  • "In function, for each input there is exactly one output."

What is odd function?

"A function f(x) is odd function if f(-x) = -f(x) "

For given question,

We know, cos(-x) = cos(x)

This means, cos function is an even function.

Consider,  f(x) = sec (5pi/3 x)

We know,

[tex]sec(\theta)=\frac{1}{cos(\theta)}[/tex]

[tex]\Rightarrow f(x) =sec (\frac{5\pi}{3x} )\\\\\Rightarrow f(x)=\frac{1}{cos(\frac{5\pi}{3x} )}\\\\\Rightarrow f(-x)=\frac{1}{cos(-\frac{5\pi}{3x} )}\\\\\Rightarrow f(-x)=\frac{1}{cos(\frac{5\pi}{3x} )}\\\\\Rightarrow f(-x)=sec(\frac{5\pi}{3x} )\\\\\Rightarrow f(-x)=-f(x)[/tex]

This means,  f(x) = sec (5pi/3 x) is an even function.

We know, sin(-x) = -sin(x)

[tex]\Rightarrow f(x)=csc(-\frac{3\pi}{2x} )\\\\\Rightarrow f(x)=\frac{1}{sin(-\frac{3\pi}{2x} )}\\\\ \Rightarrow f(-x)=\frac{1}{sin[(-\frac{3\pi}{2x} )]} \\\\\Rightarrow f(-x)=\frac{1}{-sin(-\frac{3\pi}{2x} )}\\\\\Rightarrow f(-x)=-csc(-\frac{3\pi}{2x} )}\\\\\Rightarrow f(-x)=-f(x)[/tex]

Therefore, the function [tex]f(x)=csc(-\frac{3\pi}{2x} )[/tex] is an odd function.

The correct answer is an option (B)

Learn more about odd function here:

https://brainly.com/question/14264818

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