Questions: Determine algebraically whether the given function is even, odd, or neither. f(x)=2/x^(14) Is the given function even, odd, or neither? A. Even B. Odd C. Neither

Determine algebraically whether the given function is even, odd, or neither.

f(x)=2/x^(14)

Is the given function even, odd, or neither?
A. Even
B. Odd
C. Neither
Transcript text: Determine algebraically whether the given function is even, odd, or neither. \[ f(x)=\frac{2}{x^{14}} \] Is the given function even, odd, or neither? A. Even B. Odd C. Neither
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Solution

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Solution Steps

To determine if a function is even, odd, or neither, we check the symmetry properties:

  1. A function f(x) f(x) is even if f(x)=f(x) f(-x) = f(x) for all x x .
  2. A function f(x) f(x) is odd if f(x)=f(x) f(-x) = -f(x) for all x x .

For the given function f(x)=2x14 f(x) = \frac{2}{x^{14}} , we will calculate f(x) f(-x) and compare it to f(x) f(x) .

Step 1: Determine f(x) f(-x)

Given the function f(x)=2x14 f(x) = \frac{2}{x^{14}} , we calculate f(x) f(-x) :

f(x)=2(x)14 f(-x) = \frac{2}{(-x)^{14}}

Since (x)14=x14(-x)^{14} = x^{14} (because 14 is even), we have:

f(x)=2x14=f(x) f(-x) = \frac{2}{x^{14}} = f(x)

Step 2: Check for Even Function

Since f(x)=f(x) f(-x) = f(x) , the function is even.

Final Answer

The function f(x)=2x14 f(x) = \frac{2}{x^{14}} is Even\boxed{\text{Even}}.

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